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THE ELEMENTS OF

ASTRONOMY FOR SURVEYORS.

THIRD EDITION, Thoroughly Revised and greatly Enlarged. In Crown Svo.
Pp. i-xiii-f 430. Cloth. Fully Illustrated.

A HANDBOOK ON THEODOLITE SURVEYING & LEVELLING.

For the use of Students in Land and Mine Surveying.

BY PROFESSOR JAMES PARK, F.G.S.

CONTENTS. — Scope and Object of Surveying.— Theodolite. — Chains and Steel
Bands. — Obstacles to Alignment. — Meridian and Bearings. — Theodolite Traverse.
— Co-ordinates of a Station.— Calculation of Omitted or Connecting Line in a Traverse.
— Calculation of Areas. — Subdivision of Land. — Triangulation. — Determination of
True Meridian, Latitude, and Time. — Levelling. — Railway Curves. — Mine Surveying.
— INDEX.

" A book which should prove as useful to the professional surveyor as to the
student. ' ' — Nature.

In Crown 80. Pp. i-viii + 204. Cloth. With 87 Diagrams.

PRACTICAL SURVEYING & FIELD-WORK.

Including the Mechanical Forms of Office Calculations, with
Examples Completely Worked Out.
BY VICTOR G. SALMON, M.A.,

Government Land and Mine Surveyor, Johannesburg.

CONTENTS. — Co-ordinate Calculations. — Area. — Base Measurement. — Reduction
of Field-book. — Various Problems. — Adjustment of Instruments. — INDEX.

In Crown Svo. Fully Illustrated. Cloth.

PROBLEMS IN LAND AND MINE SURVEYING.

Being 400 Questions and Answers (200 fully worked). Many Examples
taken from the Papers set by the Home Office, City and Guilds of London,
Ac., at the Surveying Examinations.

BY DANIEL DA VIES, M.I.M.E.,
County Lecturer in Mining, Surveyingj Ac.

In Cloth. Pp. i-xi + 179. Fully Illustrated.

THE EFFECTS OF ERRORS IN SURVEYING.
BY HENRY BRIGGS, M.Sc.

CONTENTS. — Introduction. — Analysis of Error. — The Best Shape of Triangles. —
Propagation of Error in Traversing. — Application of the, Methods of determining
Average Error to certain Problems in Traversing. — Propagation of Error in Minor
Triangulation. — Summary of Results. — APPENDIX.— INDEX.

" Likely to be of the highest service to surveyors . . . it is a most able treatise "
— Engineer.

FOURTEENTH EDITION, Revised. Enlarged (by 100 pages). Re-set. With
Numerous Diagrams. Cloth.

A TREATISE ON MINE-SURVEYING.

FOP the use of Managers of Mines and Collieries, Students at the
Royal School of Mines, &c.

BY BENNETT H. BROUGH, Assoc.R.S.M., F.G.S.
Revised and Enlarged by HARRY DEAN, M.SC., A.R.S.M.

CONTENTS. — General Explanations. — Measurement of Distances.— Chain Sur-
veying.— Traverse Surveying. — Variations of the Magnetic-Needle. — Loose-Needle
Traversing. — Local Variations of the Magnetic-Needle. — The German Dial. — The
Vernier Dial. — The Theodolite. — Fixed-Needle Traversing. — Surface-Surveying with
the Theodolite. — Plotting the Survey. — Plane-Table Surveying.— Calculation of
Areas. — Levelling. — Underground and Surface Surveys. — Measuring Distances by
Telescope. — Setting-out.— Mine-Surveying Problems.— Mine Plans. — Applications of
the Magnetic-Needle in Mining.— Photographic Surveying. — APPENDICES.- INDEX.

" Its CLEARNESS of STYLE, LUCIDITY of DESCRIPTION, and FULNESS of DETAIL

have long ago won for it a place unique in the literature of this branch of mining
engineering, and the present edition fully maintains the high standard of its prede-
cessors. To the student, and to the mining engineer alike, ITS VALUE is inestimable.
The illustrations arc excellent." — The Mining Journal.

London : CHARLES GRIFFIN & CO., Ltd., Exefer St., Strand, W.C.2.

PHILADELPHIA: J. B. LIPPINCOTT COMPANY.

THE ELEMENTS OF

ASTRONOMY FOR SURVEYORS

R. W. CHAPMAN, M.A., B.C.E., F.R.A.S.,

PROFESSOR OF MATHEMATICS AND MECHANICS IN THE
I'XIVERSITY OF ADELAIDE.

WITH 56 DIAGRAMS.

BH1

JSK- /^i

LONDON:

CHARLES GRIFFIN AND COMPANY, LIMITED.

PHILADELPHIA: J. B. LIPPINCOTT COMPANY.
1919.

[All Rights Resewed.}

PREFACE.

ALTHOUGH there are several excellent books on Surveying
that deal more or less thoroughly with astronomical obser-
vation, it appeared to the writer, as the result of his
experience in teaching the subject, that there is a distinct
need of an elementary work suitable for the student and
for the surveyor who is taking up astronomical observa-
tion for the first time. Most of the purely surveying
books are content to quote practical formulae for the
reduction of the observations, with little or no attempt to
expound the principles by which the formulae are derive d.
On the other hand, the theoretical works on astronomy
in which the mathematical theory is developed are gener-
ally too recondite for the beginner, and deal to a large
extent with matters of no special interest to the surveyor.
The present work is an attempt to provide an elementary
exposition, not only of the practical methods of observa-
tion and computation, but of the main principles that must
be thoroughly understood if the surveyor is to be master

405417

vi PREFACE.

of his profession. Throughout the work the methods of
observation are illustrated with numerous fully worked-out
actual observations, and a prominent feature of the book
is the attention that is given to the effects of observational
and instrumental errors of different kinds. A large pro-
portion of the examples set for working have been taken
from the papers set for candidates at the examinations for

Licensed Surveyors in Australia.

R. W. C.

ADELAIDE, September, 1918.

CONTENTS.

CHAPTER I.
THE SOLUTION OF SPHERICAL TRIANGLES.

PAGES

A Review of the Principal Formulae of Spherical Trigonometry, . 1-7

CHAPTER II.
THE CELESTIAL SPHERE AND ASTRONOMICAL CO-ORDINATES.

The Celestial Sphere — The Apparent Motion of the Stars —
Definitions of some Fundamental Terms — Astronomical Co-
ordinates— Altitude and Azimuth — Right Ascension and
Declination — Comparative Advantages of the Two Co-
ordinate Systems — The Sidereal Day and Sidereal Time —
Hour Angle — Synopsis of Astronomical Terms, . . . 8-20

CHAPTER III.
THE EARTH.

The Earth as a Globe— Terrestrial Latitude and Longitude — The
Zones of the Earth — The Altitude of the Celestial Poles equal
to the Latitude of the Place of Observation — To Find the
Shortest Distance Between Two Places whose Latitudes and
Longitudes are given — The Earth as an Oblate Spheroid —
Geographical and Geocentric Latitude — Examples, . . 21-34

CHAPTER IV.

THE SUN.

The Sun's Apparent Motion among the Stars — The Earth's Orbit
Round the Sun — The Equinoxes — The Sun's Motion in Right
Ascension and Declination — The Sun's Semi-Diameter —
Plotting the Position of the Sun's Centre on the Celestial
Sphere — The Sun's Apparent Annual Path, .... 35-43

viii CONTENTS.

CHAPTER V.

TIME.

PAGES

Sidereal Time — Apparent Solar Time — Mean Time — The Three
Systems of Time Measurement — Equation of Time — Local
Mean Time — Local Sidereal Time — Standard Time — To
Change Standard Time to Local Mean Time— To Reduce an
Interval of Mean Time to Sidereal Time, and vice versa — To
Find Local Sidereal Time at Local Mean Noon, given the
Sidereal Time at Mean Noon at Greenwich — Given the
Local Mean Time at any instant to Determine the Local
Sidereal Time — Given the Sidereal Time to Find the Corre-
sponding Local Mean Time — Alternative Methods for
Preceding Problems — Determination of Time of Transit of a
known Star across the Meridian — Time of Transit of the First
Point of Aries — The Use of the Greenwich Time of Transit
of the First Point of Aries in Computations of Local Mean
and Sidereal Time — Nautical Almanac Data with Regard to
Time Calculations — Examples, ...... 44-70

CHAPTER VI.
THE LOCATION OF OBJECTS ON THE CELESTIAL SPHERE.

Given the Right Ascension and Declination of a Star, to De-
termine its Altitude and Azimuth at any Time — Having
Observed the Altitude and Azimuth of a Star, and Noted the
Time, to Compute its Right Ascension and Declination —
Having Determined the Altitude aud Azimuth at a given
Time, to Find the Approximate Position of the Star at some
Short Interval of Time afterwards — Examples, . . . 71-79

CHAPTER VII.

ASTRONOMICAL AND INSTRUMENTAL CORRECTIONS TO OBSERVATIONS
OF ALTITUDE AND AZIMUTH.

Parallax— Horizontal Parallax — Atmospheric Refraction — Correc-
tions to Observations on Account of Residual Instrumental
Errors — The Effect of an -Error in Collimation — The Elimina-
tion of Instrumental Errors by Changing Face — The Error
made if the Transverse Axis of the Telescope is not Hori-
zontal— The Use of the Striding Level — Allowance for Error
of Alidade Level 80-96

CONTENTS. ix

CHAPTER VIII.

THK DETERMINATION OF TRUE MERIDIAN.

PAGES

Referring Marks — First Method : By Equal Altitudes of a Cir-
cumpolar Star. Second Method : By a Circumpolar Star at
Elongation — Calculation of the Time of Elongation — The
Effect of an Error in Latitude — Star Observations in Day-
light. Third Method : By Extra-Meridian Observations on
Sun or Star — SUD Observations — Computation of Sun's De-
clination from Nautical Almanac Data — The Effects of Errors
in Latitude, Declination and Longitude. Fourth Method :
By Time Observations upon a Close Circumpolar Star —
Circum- Elongation Observations for Azimuth — Examples, - 97-148

CHAPTER IX.
THE DETERMINATION OF LATITUDE.

First Method : By Meridian Transits — Zenith Pair Observations
of Stars — Meridian Altitudes at both Lower and Upper
Culminations. Second Method : By Circum-Meridian Obser-
vations— Circum-Meridian Observations of the Sun. Third
Method : By Prime Vertical Transits — The Effects of Errors
in the Time Measurement and in the Setting Out of the
Prime Vertical — Striding Level Correction. Fourth Method :
By the Altitude of the Pole Star at any Time — A Rough
Method by Noting the Rate at which Altitude of Sun or Star
changes near the Prime Vertical — A Method by the Measure-
ment of the Horizontal Angle between Two Circumpolar
Stars at their Greatest Elongations.— Table for Reduction of
Circum-Meridian Observations - Examples, . . . .149-181

CHAPTER X.
THE DETERMINATION OF TIME BY OBSERVATION.

First Method : By Meridian Transits— The Effect of an Error in
the Direction of the Meridian — The Effect of an Error in the
Horizontality of the Transverse Axis — Meridian Transits on
both Sides of the Zenith. Second Method : By Extra-
Meridian Observations of Sun or Star — Arrangement of the
Computation — Averaging several Observations— Observations
on both East and West Stars— The Effect of Errors in
Latitude, Declination and the Measured Altitude. Third

CONTENTS.

I'AGKSr

Method : By Equal Altitudes— Error Due; to Inequality in
the Altitudes — Application of Method to Sun Observations.
Fourth Method: Almucantar Method for Time Observations-
Sun Dials — The Horizontal Dial — Prime Vertical Dial
Oblique Dials — Time of Rising or Setting of a Celestial Body
—Examples, ". 182-21S

CHAPTER XL
THE DETERMINATION or LONGITUDE.

1. By Portable Chronometers. 2. By Electric Telegraph or Wire-
less Telegraphy — Recording and Receiving Signals — Com-
parison of Chronometers — Personal Equation — Programme of
Operations. 3. By Flash Lamp Signals— Longitude by Lunar
Observations— (a) By Lunar Distances— (b) By Lunar Cul-
minations—(c) By Lunar Occultations, . . . .219-235

CHAPTER XII.

THE CONVERGENCE OF MERIDIANS, ..... .236-241

INDEX, . . . .243-247

ASTRONOMY FOR SURVEYORS.

CHAPTER I.
THE SOLUTION OF SPHERICAL TRIANGLES.

IN this chapter the principal formulae of spherical
trigonometry, such as will be afterwards applied to
calculations on the celestial sphere, are brought together
for convenient reference. No attempt will be made to
establish the formulae, which are demonstrated in any of
the ordinary books on spherical trigonometry, but a brief
synopsis will be given of the usual methods for the
solution of spherical triangles under different conditions.

Great Circles. — The line of intersection made with the
surface of a sphere by a plane passing through the centre
of the sphere is known as a great circle. If this circle
passes through two points A and B on the surface of the
sphere, then the shortest distance between A and B,
measured along the sphere's surface, is that measured
along the arc of the great circle joining them. Only one
great circle can be drawn to pass through two given
points on the surface of a sphere, unless they happen
to be at opposite extremities of a diameter, and the
length of the shorter arc of this great circle between the
two points is the shortest distance between them. Meri-
dians of longitude on the earth's surface are great circles.

In spherical trigonometry it is always assumed that
the arcs representing the sides of the triangles considered
are arcs of great circles.

2 ASTRONOMY^ FOR SURVEYORS.

Small Cfrcltts:-^Tne line ' of l intersection made with the
surface of a sphere by a plane that does not pass through
the centre is known as a small circle. The ordinary
formulae of spherical trigonometry do not apply to tri-
angles having sides that are arcs of small circles. A
parallel of latitude on the earth's surface is a small circle.
It follows that the shortest distance between two points
in the same latitude is not that measured along the
parallel of latitude, but is measured along the arc of the
great circle joining them.

Spherical Triangles. — Denoting the angles of a spherical
triangle by A, B, and C, and the sides opposite to these
angles by a, b, and c respectively, the sides being as
usual measured by the angles which they subtend at the
centre of the sphere, then we have the following funda-
mental relations : —

(a) The sines of the angles are proportional to the
sines of the opposite sides : —

sin A _ sin B _ sin C
sin a sin b sin c

(b) One side of a triangle is expressed in terms of the
two other sides, and the angle included between them by
one of the three formulae : —

cos a = cos b cos c + sin b sin c cos A\

cos b = cos c cos a + sin c sin a cos B > . (2)

cos c = cos a cos b + sin a sin b cos C;

cot a sin b = cot A sin C + cos 6 cos C^

• ( O }

cot b sin a = cot B sin C -f- cos a cos Cj v

Whilst the formulae (2) and (3) are extremely useful in
all sorts of investigations into the properties of spherical
triangles, they are not adapted to logarithmic computation,

THE SOLUTION OF SPHERICAL TRIANGLES. 3

and are consequently not suitable for use in the numerical
solution of triangles. For this purpose other formulae are
commonly used, derived from these fundamental relation-
ships but expressed in a form more suitable for use with
logarithms.

The Solution of Right- Ar.gled Spherical Triangles.— The
relationships between the sides and angles of a right-
angled spherical triangle are very conveniently summarised
by the mnemonic rules due to Napier, the inventor of
logarithms, and known as Napier's Rules of Circular
Parts.

Denoting the right angle by C, Napier defines five
"circular" parts (i.e., a, 6, 90°— A, 90° —c, 90° — B),
and these are supposed, as in
the figure, to be ranged round
a circle in the order in which
they stand in the triangle.
Then, if any one of these five
parts is selected and called the
middle part, the two parts on
each side of it are called the
adjacent parts, and the remain-
ing two are called trie opposite
parts. For instance, if a is chosen as the middle part,
90°— B and b are the adjacent parts,, and 90°— c and
90° — A are the opposite parts. Then Napier's Rules
are : —

Sine of middle part ^product of tangents of adjacent
parts.

Sine of middle part =product of cosines of opposite

parts.
Thus

sin a =cot B tan b
and sin a =sin c sin A.

4 ASTRONOMY FOR SURVEYORS.

As an aid to memory, it may be noticed that the vowels
in the words sine and middle are the same, so with tangent
and adjacent, cosine and opposite.

By choosing different parts in turn as the middle parts,
we obtain all the possible relationships between the sides
and angles, and with a little practice it is easy to choose
the particular ones wanted. If we want a relationship
between a, 6, and c, for example, 90°— c must be taken
as the middle part, and we have

cos c=cos a cos 6.

[f a relationship between a, A, and B is required, take
90°— A as the middle part, whence

cos A =sin B cos a

and so on.

There are six cases to consider in the solution of right-
angled triangles, and the formulae required, readily
obtained from Napier's rules, are as follows : —

(1) Given the hypotenuse c and an angle A.

tan b = tan c cos A,
cot B = cos c tan A,
sin a --= sin c sin A.

(2) Given a side b and the adjacent angle A.

tan b

tan c =

cos A'

tan a = tan A sin b,
cos B = cos b sin A.

(3) Given the two sides a and b.

cos c = cos a cos b,
cot A= cot a sin b,
cot B = cot b sin a .

THE SOLUTION OF SPHERICAL TRIANGLES, 5

(4) Given the hypotenuse c and side a,

cos c

cos o = ,

cos a

tan a

cos B = - — ,
tan c

sin a

sin A= . — .
sin c

(5) Given the two angles A and B,

cos c = cot A cot B,

cos A

cos a = -.-• -,
sin B

cos B

cos o = •- -.

sin A

(6) Given a side a and opposite angle A,

sin a

sin c = - — ,
sin A

sin b = tan a cot A,

cos A

sin B = — .
cos a

The Solution of Oblique Spherical Triangles. — (1) Given
the three sides, a, b, and c.

Let s=l (a +6+ c).

Then the angle A may be computed from any one of
the following three formulae :—

A / sin (s — b) . sin (s — c)
sin - A/ - L-^-TL:-T-
2 sin b . sin c

A /sin s . sin (s — a)

2 * sin b . sin c

/sin (5 — 6) . sin (•<? — c}
tanA=y — r- -V- -- ~ •

sm ,s . sin «s—

6 ASTRONOMY FOR SURVEYORS.

Similar formulae apply, of course, to the other two
angles.

(2) Given two sides a and 6, and the included angle C,

cos

sin \

These determine J(A+B) and |(A— B), and hence,
by addition and subtraction, A and B.
c may either be found from

sin a sin C

sin c =

sin A

The former of the two alternative formulae for c is the
simpler, but as the value of c is here found from its sine,
it is sometimes difficult to determine which of two values
is to be given to it. This difficulty does not arise with
the second formula.

(3) Given one side c and two adjacent angles A and B.

C°S * ~

cos J (A-f-B)
sin \ (A-B)

tan c.

These determine J (a + 6) and \ (a~b), and hence, by
addition and subtraction, a and 6.
C may be found either from

sin A . sin c

sinC =

sin a

sm J (a

THE SOLUTION OF SPHERICAL TRIANGLES. 7

Similar remarks applying to the two formulae as in
case (2).

(4) Given two sides a and 6, and the angle opposite
one of them A.

This is generally known as the ambiguous case.

B may be found from

sin 6 ,

sin B = - - sin A,
sin a

which will usually determine two possible values of B.
If the value of sin B obtained is greater than unity there
will be no solution at all.

Having determined B, C and c may be found from the
formulae : —

(5) Given two angles A and B, and the side opposite
one of them, a.

The solution in this case is similar to case (4), and two
solutions are often possible :—

sin B sin a

sin 6 = -- : — - - ,
sin A

after which the same two formulae as in case (4) determine
tan | C and tan \ c.

Spherical Excess.— -The sum of the three angles of a
spherical triangle is always greater than 180°, the differ-
ence A + B + C — 7t being known as the spherical excess.

If this is denoted by E, the area of any spherical triangle
— E r2, the spherical excess being in circular measure,
and r denoting the radius of the sphere.

CHAPTER II.

THE CELESTIAL SPHERE AND ASTRONOMICAL
CO-ORDINATES.

The Celestial Sphere. — We may easily imagine, looking
up to the heavens on a cloudless night, that the stars
are distributed over the surface of the spherical vault
of sky above us. It is not really so, because refined
measurements have proved that the distances of the
stars differ tremendously, but these distances are so
immense that in most cases they cannot be measured
even by the most skilful of astronomers with the most
delicate of instruments. The consequence is that for
practical purposes we are never concerned with the
distances of the stars, but only with their directions,
and in order to record these it is exceedingly convenient
to picture the stars as distributed over the surface of an
imaginary spherical sky having its centre at the position
of the observer. Thus has arisen the conception of the
Celestial Sphere, which we may consider as a geometrical
device to enable us to record and measure the directions
of the stars.

In Fig. 1, suppose that 0 represents the position of
the observer. With O as centre, imagine a spherical
surface described with a radius of any length we please ;
we may make it a few feet or a few thousand miles, it
makes no difference. Now, let A, B, and C be three of
these immensely distant stars, and let the lines 0 A, OB,
and 0 C cut our imaginary sphere in a, 6, and c respec-
tively. Then, if we are only concerned with the directions
of the stars, we may just as well picture them as occupying
the positions a, b, and c as their actual places A, B, and C.

CELESTIAL SPHERE AND CO-ORDINATES. 9

In fact, to the observer at O their appearance would be
unaltered. So, proceeding in this way, we may picture
all the stars in the sky as occupying places on this imagi-
nary surface, which is then known as the Celestial Sphere.
It may be considered as the spherical surface upon which
the stars appear to lie, but, of course, in reality they are
not all equally distant from us, and they are only repre-
sented in this way in order to conveniently measure their
directions. ^

If through the point O a vertical line be drawn to

intersect the celestial sphere over the observer's head
-ku_Z; and to cut it vertically below his feet at N, the
point Z is called the Zenith and the point N the Nadir.
The Zenith is thus the point in the celestial sphere directly
over the observer,

If a horizontal plane H R be drawn through O, a plane
— that is to say, at right angles to the vertical O Z, the
direction in which gravity acts — it will cut the celestial

10 ASTRONOMY FOR SURVEYORS.

sphere in a great circle, which is called the Celestial
Horizon. To an observer whose eye was close to the
surface of a calm ocean, the celestial horizon would form
the boundary of the visible part of the celestial sphere.

The Apparent Motion of the Stars. — Continued observa-
tion shows that, leaving the few planets out of account,
the other stars always maintain the same relative posi-
tions, and hence they are commonly referred to as the
fixed stars. Whilst, however, there is no motion relative
to one another, they all appear to revolve from East
to West in a period slightly less than twenty-four hours
round a point in the sky that is known as the celestial
pole. The motion is just as though the whole celestial
sphere, carrying the stars, revolved about an axis passing
through this point and its own centre. The ancients,
who regarded the earth as a flat plane, thought that
this was really what occurred, but we know now that
this motion is apparent only, and is due to the fact that
we view the stars from a revolving earth. Thus, referring
to Fig. 2, the whole of the stars appear to slowly describe
circles about a point P in the celestial sphere just as
though the whole sphere revolved about the axis O P,
so that every star completes its circle in the same time.
Some stars, such as A, which are comparatively near to
the point P, describe only a small circle, which never
takes them below the horizon, so that such stars are
always visible. Thus the Southern Cross in the latitude
of Southern Australia can be seen at all times, and never
sets. Other stars, such as B and C, which are further
away from P, describe much larger circles, which take
them, as is shown in the figure, below the horizon for a
portion of their revolution, so that such stars rise in
the East and set in the West. This diurnal motion of
the stars may be very prettily demonstrated by exposing
a fixed camera containing a highly sensitised plate directed
towards the celestial pole on a clear night, leaving the

CELESTIAL SPHEEE AND CO-ORDINATES.

plate exposed for an hour or two. The images of the
brightest moving stars will leave trails upon the plate
which are all seen to be arcs of circles having a common
centre at the celestial pole.

Now, the stars are so distant that their apparent
direction in space is absolutely unaltered by any move-
ment of the observer over the earth's surface. The
direction of any particular star is precisely the same,
even when determined by our most refined instruments,
whether viewed from Melbourne, London, or Perth.

Fig. 2.

More than this, we know that the earth, in the course
of a year, describes a path round the sun that is approxi-
mately a circle whose diameter is over 190 millions of
miles, yet even this great shift of the point of observation
produces no appreciable change in the directions of the
fixed stars. At intervals of six months apart, when the
points of observation, that is to say, are distant something
like 190 millions of miles, a slight difference in direction,
amounting to only a fraction of a second of arc, may
be detected in a few stars with the refined observations

12 ASTRONOMY FOR SURVEYORS.

possible at fixed observatories. But even this cannot be
found with the great majority of the stars, so that
we may regard the position of the observer on the earth's
surface as of absolutely no importance when measuring
the direction of the stars in space. Looking at Fig. 2,
we may regard the earth as a tiny speck at O, the centre
of the great celestial sphere, and no matter where we
take the point 0 on this tiny speck, the direction of the
line O P remains the same within the possibilities of our
means of measurement, so that the lines joining any one
of the fixed stars to different points on the earth's surface
may all be considered as parallel.

It follows from this that the portion of the sky visible
to an observer at any point on the earth's surface presents
exactly the same appearance as it would do if it were possible
for him to view it from the earth's centre. This statement
refers only to the fixed stars.

Therefore, if we imagine an observer anywhere on a
small spherical earth at the centre of a great celestial
sphere of dimensions indefinitely great compared to the
earth, and suppose the earth to rotate about an axis
through its centre, the successive pictures of the sky
presented to the observer during a revolution will be
precisely the same as they would be if the earth remained
stationary and the great celestial sphere itself were to
rotate about the same axis.

Thus, looking at Fig. 2, if we produce the line P 0
backwards to cut the celestial sphere below the plane
of the horizon in P1, the fixed stars appear to the observer
at O to revolve on the celestial sphere about the axis
P P1. In reality it is the earth that is revolving, and it
is the earth's axis that lies in the direction P P1, so that
the celestial poles P and P1 are the points in which the
axis of the earth, if indefinitely produced, would cut
the celestial sphere. If the observer is in the Southern
Hemisphere, the pole P visible to him will be that to which

CELESTIAL SPHERE AND CO-ORDINATES. 13

the earth's South Pole is directed. If he is in the Northern
Hemisphere the visible celestial pole is that towards
which the earth's North Pole points. j$& ^

Celestial Equator. — If we take a plane through O **l
perpendicular to the line^? P1, it will cut the celestial
sphere in a great circle Ifcft, which is known as the Celestial , /
equator. Its plane clearly is coincident with the plane ,*
of the equator of the earth. Since two great circles of a T^
sphere always intersect at opposite extremities of a dia-
meter, it follows that a star revolving in the celestial }*
equator has its path divided into two equal parts by the
circle of the celestial horizon H R, so that the time during
which it is visible above the horizon will be equal to the
time it is out of sight below.

Thus, to an observer in Southern latitudes, the celestial \
pole P lies to the south and, since the line P P1 (Fig. 2)
marks also the direction of the earth's axis, the celestial
pole will be in the direction of the true geographical
South. Any star, such as B, lying to the South of the
celestial equator, will trace the greater part of its circular
path above the plane of the horizon. On the other hand,
a star, such as D, to the North of the celestial equator,
will trace out the smaller portion of its path only above
the horizon, so that it will be visible for less than half
of its time of revolution. Stars such as E, sufficient^
far to the North, will not be visible at all to a person in
this latitude, but will complete the whole of their revolu-
tion below the plane of the horizon, as shown in the
figure.

Astronomical Co-ordinates — If we wish to mark the
position of a point on a plane, we may do so by measuring
its distances from two fixed straight lines at right angles.
A knowledge of these two distances is sufficient to enable
us to fix the position of the point, but one distance only
would not be enough. Measured in this way, these two
distances are spoken of as the " co-ordinates " of the

ASTRONOMY FOR SURVEYORS.

point. Now, in astronomical observation, we commonly
require to determine the position of a star on the celestial
sphere, and so it is necessary to have some system of
co-ordinate measurement applicable to the purpose.
Either one of two sets of co-ordinates is commonly em-
ployed. The first set is Altitude and Azimuth.

In Fig. 3, let O be the position of the observer, Z the
zenith, P the celestial pole. Then the plane Z 0 P will
cut the plane of the horizon through 0 in the North and
South points N and S. S Z N is known as the plane of
the Meridian.

Suppose that B is a star describing its circular path
ABC round the pole P.

Fig. 3.

The plane Z B O cuts the plane of the horizon in the
line D O.

Then it is clear that if we know the angle DON, which
is the angle that the plane Z 0 D makes with the plane
f of the meridian, our knowledge is sufficient to fix the
I position of the plane Z 0 D.

If in addition we know the angle BOD, the position
of the star B may be fixed on the celestial sphere.

The angle. DON, which the plane passing through the
zenith and the star makes with the meridian, measures
what is known as the azimuth of the star. It is generally
measured from the North towards the right.

CELESTIAL SPHERE AND CO-ORDINATES. 15

The angle BOD, measuring the angular altitude of
the star in a vertical plane above the horizon, is spoken
of as the altitude of the star.

Instead of the altitude we may measure the angle
Z 0 B, which is known as the Zenith Distance, and is
clearly the complement of the altitude.

If we know both the altitude and azimuth of a star
at any time we can mark its position on the celestial
sphere. The ordinary theodolite is adapted for measure-
ment in this system of co-ordinates.

The second or alternative set of co-ordinates is Right
Ascension and Declination.

In ,Fig. 4, let 0 be the position of the observer, Z the
zenith, P the celestial pole, and S P Z N the plane of the
meridian.

Suppose that B is a star travelling round the pole in
the direction of the arrow in a circle of which only half
is shown.

Q D Q' is the plane of the celestial equator drawn
through O at right angles to 0 P.

P B D is the arc of a great circle of the celestial sphere
intersecting the celestial equator in D. The plane of this
great circle must pass through O, and the angle P O D is
a right angle.

Then clearly if we know the position of the point D
on the celestial equator, and also know either the angle
P 0 B or the complementary angle BOD, we shall be
able to fix the position of the star B on the celestial
sphere.

The position of the point D on the equator may be
determined if we know its angular distance from some
known fixed point also on the equator. The fixed point
selected for the purpose is known as the First Point of
Aries. It is usually indicated by the symbol <Y> , denoting
a pair of ram's horns. The exact nature of this point
we shall discuss a little later on, but for the present all

ASTRONOMY FOR SURVEYORS.

that we want to know is that it is a point whose position
'can always be accurately determined.

If we know, then, the angular measure of the arc v D
— that is to say, the angle which the arc subtends at the
centre 0, and also the direction in which it is measured
from <v» — that is sufficient to determine D.

To avoid any confusion as to the direction in which
the arc <Y» D should be measured, it is always measured
from <Y> towards the East — that is to say, in the opposite
direction to that in which <v» travels round the celestial

Fig. 4.

equator Q Q' — because on moves round with the rest
of the fixed stars from East to West.

Measured in this way, the angular measure of the
arc <v» D is known as the Right Ascension .of the star B.
It may have any value from 0° to 360°. It is commonly
denoted by the letters R.A.

The Right Ascension of the star being known, its
position may be fixed if we know either the angle FOB,
the angular measure of the arc P B, or the angle DOB,
the angular measure of the arc D B.

CELESTIAL SPHERE AND CO-ORDINATES. 17

The angular measure of the arc P B is known as the
Polar Distance of the star B. It is generally denoted by
the letters N.P.D. or S.P.D., according as it is measured
from the North or the South Pole.

The angular measure of the arc D B is called the
Declination of the star B, and the circle P B D is known
as the Declination Circle of the star. The declination is
said to be North or South according as the star is North
or South of the*eq[u'alor.

Polar Distance and Declination are always comple-
mentary to one another, their sum being 90°, so that
if one is known the other is found by simple subtraction
from 90°.

Comparative Advantages of the two Co-ordinate Systems.

-The altitude and azimuth of a star are readily measured

with a theodolite, and serve to fix the position of a star

at any particular instant, but owing to the diurnal motion

of the stars these co-ordinates are continually changing.

On the other hand, the right ascension and declination
of a star are constant, for the reference point, the first
point of Aries, partakes of the diurnal motion of the stars.
These co-ordinates are in consequence the most convenient
for recording the relative positions of the stars on the
celestial sphere. Thus in the Nautical Almanac the stars
are catalogued and tabulated by their right ascensions
and declinations.

The Sidereal Day and Sidereal Time. — As the revolution
of the whole system of stars about the polar axis takes
place with absolute uniformity from East to West, the
period of revolution serves as a convenient unit of time
for astronomical purposes. All the stars complete their
circles of revolution in the same period, which is known
as the sidereal day. This day is about 4 minutes shorter
than the ordinary day. Sidereal clocks, adjusted to keep
sidereal time, the sidereal day being divided into 24 hours,
are used in fixed observatories. Such clocks are arranged

18 ASTRONOMY FOR SURVEYORS.

to mark 0 hr. 0 min. 0 sec. when the first point of Aries,
the point on the celestial equator from which Right
Ascensions are measured, crosses the meridian of the
observer. Thus the sidereal time at any instant is the
interval that has elapsed, measured in sidereal hours,
minutes, and seconds, since the last transit across the
meridian of the first point of Aries.

Looking at Fig. 4, it is clear that all stars on the same
declination circle, such as P B D — that is to say, all
stars having the same right ascension — will cross the
meridian at the same instant. A star whose right ascen-
sion is 180° will cross the meridian 12 sidereal hours after
the first point of Aries, and one whose right ascension
is 15° will cross the meridian at 1 hr., sidereal time. Thus
we deduce the important result that the right ascension
of a star, when reduced to time at the rate of 24 hours for
360° or 1 hour for 15°, gives the sidereal time at the moment
when it crosses the meridian.

—> »»«

Hour Angle. — In Fig. 4 the angle R^P^-^whMir-is-the
angle that the plane of the declination circle P B D makes
with the plane of the meridian, is known as the hour
angle of the star B. If we know the hour angle of a star,
and also its polar distance, we can clearly mark the
position of the star on the celestial globe, so that these
two may be used as another system of co-ordinates. The
hour angle of a star is continually changing, but owing
to the uniform character of the star's motion, it varies
«.f p pppafg/nf. raff j If the hour angle is 90° measured
towards the East, then the star will take 6 sidereal hours
to reach the meridian. Thus a knowledge of the hour,
angle at once gives us the time the star will take to reach
the meridian, if it be on the East side of it, or the time
that has elapsed since the star crossed the meridian, if
it be on the Western side.

Prime Vertical. — The plane through the zenith at right
angles to the meridian — that is, the vertical plane running

CELESTIAL StflERE AND CO-ORDINATES. 19

East and West — is Is
East and West line,1

lown as the Prime Vertical. The
>yhich is the line of intersection of

th the plane of the horizon, is also
of the plane of the celestial equator
ill be evident from Fig. 2.
al Terms. — For purposes of reference,
es dealt with in this chapter are
e.

the Prime Vertical w
the line of mfefsectidi
with the horizon, as j\

Synopsis of Astronomi
the principal quantit
illustrated in one figu

Fig. 5a is drawn for an observer in the Southern Hemi
sphere, and Fig. 5b for the Northern Hemisphere.

EXAMPLES.

1 . The R.A. of a star being 35° 20', what is the local sidereal time when
the star is in the meridian ?

Ans. 2 hrs. 21 min. 20 sec.

2. If the R.A. of a star is 295° and the sidereal time is 15 hours, is the
star to the East or West of the Meridian ?

Ans. To the East.

3. What is the declination of a star that rises exactly in the East ?

Ans. 0°.

4. What is the hour angle of the star in Question 2 ?

Ans. 70°.

5. The declination of a star is 35° South. Determine its S.P.D. and its
N.P.D.

Ans. 55° and 125°.

6. If the First Point of Aries crosses the meridian exactly two hours,
as measured by a sidereal clock, after a certain star, what is the R.A. of
the star ?

Ans. 330°.

7. The declination of the Pole Star is 88° 51' North. What is the difference
between its greatest and least zenith distances ?

Ans. 2° 18'.

8. At the time of the year when the R.A. of the sun is zero, determine
approximately the time of rising of a star with declination 0° and R.A.
150°.

Ans. 4 p.m.

9. What is the point whose altitude is 90° and hour angle zero ?

Ans. The zenith.

ASTRONOMY FOR SURVEYORS

A I

Fier. 5«.

Fig. 56.

O is the observer.

S W N E, the plane of the horizon.

Z, the zenith.

P, the celestial pole ; 0 P, the

polar axis.

S P Z N, the plane of the meridjan.
K' W Q E, the celestial equator.
W Z E, the prime vertical.
N, S, W, K, the North, .South,

West, and East points.

B, any star.

Z P B, the hour angle of B.

P B D, the declination circle of B.

P B, the polar distance of 1>.

B D, the declination of B.

of> U, the right ascension of B.

Z B P, the vertical through B.

BF, the altitude of B.

B Z, the zenith distance of B.

N F, the azimuth of B.

CHAPTER III.

THE EARTH.

The Earth a Globe. — That the earth is a globe is no longer
a matter for dispute. It has been circumnavigated and
.mapped and measured, and no other supposition will
fit the facts. We see its round shadow as cast upon the
moon during a partial eclipse. We see the planets as
great balls of similar dimensions revolving at different
distances round the great central sun. The law of gravi-
tation explains the form of their orbits and enables their
movements to be predicted with the greatest exactness.
That our earth is a globe like these, revolving in a similar
way around the sun, is the only satisfactory hypothesis
that will account for their apparently involved move-
ments in the heavens. The whole of the apparent move-
ments of the heavenly bodies are readily accounted for
on the supposition that the earth is a globe, and no
explanation even plausibly satisfactory has been advanced
on any other supposition.

In the case of some of the planets we can actually
observe that they are in rotation in a manner similar to
that in which we assume our own earth must rotate to
account for the phenomena of night and day and of the
diurnal rotation of the stars. In the planet Mars we see
the poles or extremities of the axis of rotation surrounded
by white caps apparently similar to the great caps of ice
and snow that surround the poles of our own earth.

Terrestrial Latitude and Longitude. — The extremities of the
axis of rotation of the earth are called the Poles, and are
distinguished as the North and South Poles.

ASTRONOMY FOR SURVEYORS.

A plane through the earth's centre at right angles to
the axis cuts the earth's surface in a circle known as the
Equator. Every point on the terrestrial equator is thus
equidistant from the North and South Poles.

In order to mark the position of a point on the earth's
surface, it is necessary to have a system of co-ordinates
similar to those we have already discussed in connection
with the celestial sphere.

Suppose that P (Fig. 6) is a point on the earth's surface,
the position of which it is desired to locate. A plane

Eig. 0.

passing through P and the earth's axis will cut the earth's
surface in a great circle N P M S, which is known as a
Meridian. Suppose this Meridian cuts the equator at
the point M. Then clearly, if we know the position of
the point M on the equator, and also the length of the
arc P M or the angle which it subtends at the earth's
centre, we shall be able to fix the point P.

The position of M on the equator is determined by the
longitude of P.

To measure this, some arbitrary place A must be

THE EARTH. 23

selected on the equator as a starting point. The point
actually chosen is the point of intersection of the meridian
passing through Greenwich, shown as N G A S in the
figure, and the equator. The angular measure of the arc
A M— that is to say, the angle A 0 M — is known, as the
longitude of P. Thus, all points on the meridian passing
through P have the same longitude. All points on the
meridian N G A S, passing through Greenwich, have zero
longitude. The longitude of other places is reckoned as
so many degrees East or West of Greenwich until we
come, to 180°, which is the longitude of the meridian
exactly opposite to the Greenwich meridian.

The angle POM, which is the angle between the
direction of the vertical at P and the vertical at M,
measures what is known as the latitude of P. If we draw
a plane through P at right angles to the earth's axis,
it will intersect the earth in a small circle L P L' parallel
to the equator. Such a circle is called a Parallel of Lati-
tude, and all points on the same parallel clearly have the
same latitude.

Latitude is measured as so many degrees North or
South of the Equator. The latitude of the North Pole
is 90° N.

Thus, if we know the position of the meridian of 'zero
longitude, the latitude and longitude of a place are suffi-
cient to enable us to mark its position on the globe.

The Length of a Degree of Longitude. — If the parallel
of latitude through P intersects the meridian through
Greenwich in B, it is clear that the arc B P will be much
smaller than the arc AM. It will have the same angular
measurement on a much smaller circle. If P were very
near to the North Pole, the arc B P would be very small
indeed. Thus two places in the same latitude but differing
by, say, ten degrees of longitude, will be very much closer
together if they are in a " high " latitude — that is to say,
a latitude approaching 90° — than they will be if both

ASTRONOMY FOR SURVEYORS.

are on or near the equator. Thus a degree of longitude
has its greatest value, when measured in distance along
the earth's surface, at the equator, its value becoming
less and less as we approach the poles. At the equator
a degree of longitude is equivalent to a distance of about
69 miles.

A degree of latitude, on the other hand, is always of
approximately the same value, about 69 miles, whether
it is measured near the poles or near the equator, because
it is measured along meridians which are all great circles
of the same diameter.

The Zones of the Earth. — Certain parallels of latitude

Arctic Circle

Tropic of Cancer.

Equator.

Tropic of. Capri

divide the earth's surface into five belts or divisions,
termed zones. These mark in a general way a natural
division of the earth's surface according to climate. The
parallel of latitude 23° 27J' North of the Equator is
termed The Tropic of Cancer, and the corresponding
parallel South of the Equator is termed The Tropic of
Capricorn. As we shall presently see, at all places between

THE EAETH. 25

these parallels at some part of the year the sun shines
directly overhead at mid-day. As a consequence, the belt
included between these is the hottest portion of the
earth's surface, and it is known as the Torrid Zone.

The parallel of latitude 66° 32J' North of the Equator
is called the Arctic Circle, and the corresponding parallel
South of the Equator the Antarctic Circle. The belt
between the Arctic Circle and the Tropic of Cancer is
known as the North Temperate Zone, and that between
the Antarctic Circle and the Tropic of Capricorn as the
South Temperate Zone. The regions around the two poles
bounded by the Arctic and Antarctic circles respectively
are termed the Frigid Zones. At all places within the
frigid zones the sun is below the horizon at mid-day for
some portion of the year.

The Altitude of the Celestial Pole is Equal to the Latitude of the
Place of Observation. — In Fig. 8, let O be the position of the
observer and C the earth's centre. Then the direction of
the pull of gravity at O is in the direction O C. This,
then, will mark the direction of the vertical at O,
and the zenith, Z, of the observer will be in C O
produced.

H R. at right angles to O Z, marks the plane of "this
horizon.

If C P, the earth's axis, be produced to cut the celestial
sphere in P1; then Px will be the celestial pole.

Draw O P2 parallel to C Px.

Then the celestial pole being, as we have seen, at ax
distance from the earth that is practically infinite in
comparison to the earth's radius, 0 P2 wjll mark the
direction in which the celestial pole is seen by the observer
at O.

Draw the plane of the equator E C Q at right angles
to the earth's axis.

Then, from our definition, the latitude of 0 is measured
by the angle ECO.

26 ASTRONOMY FOR SURVEYORS.

Now the angle ZOP2=the angle O C P1? and the
complements of these angles are equal.

Therefore, the angle P2OR=the angle E C 0— i.e.,
the altitude of the pole = the latitude of the observer.

It follows from this that if the observer travels equal
distances North and South from 0, since his latitude will
change by equal amounts, the altitude of the celestial
pole will also be increased or decreased by equal amounts.
As this is actually the case from observation, the fact
forms a strong proof of the sphericity of the earth.

To find the Shortest Distance, measured along the Earth's
Surface, between two Places whose Latitude and Longitude are
given, assuming the Earth to be a True Sphere.

In Fig. 9, let P and R be two places whose latitudes
and longitudes are known.

The shortest distance between P and R, measured
along the earth's surface, will be the length of the arc
of the great circle joining them.

THE EARTH.

Draw the meridians passing through P and R.

Then if we know the latitudes, we know the angular
measure of the meridian arc* N P and N R, N being the
North Pole.

If P is in North latitude, the arc N P is the complement
of the latitude. If R is the South latitude, the arc N R
is 90°+ the latitude.

The angle P N R is the difference of the longitudes of
P and R if both are measured in the same direction, or

Fig. 9.

the sum of the longitudes, if one is East and the other
West.

Thus in the spherical triangle N P R, we know the sides
N P and N R and the included angle P N R.

Then by the ordinary methods of spherical trigonometry
we can compute the angular measurement of the great
circle arc P R, and consequently its lineal measurement,
if we know the radius of the earth.

The radius of the earth is approximately 3,960 miles.

ASTRONOMY FOR SURVEYORS.

EXAMPLE. — Find the shortest distance measured along
the earth's surface between Perth (long. 115° 50' E., lat.
31° 57' 8.) and Brisbane (long. 153° I7 E., lat. 27° 28' S.)9
assuming that the earth is a sphere of radius 3,960 miles.

In this case, both places being in the Southern Hemisphere, it will be
preferable to solve the triangle S P R (Fig. 9) rather than N P R.

If A denotes the position of Brisbane, B of Perth, and C the South Pole,
we shall have in the spherical triangle ABC

C A = b = 90° - 27° 28' = 62° 32'

C B = a = 90° - 31° 57' = 58° 03'

C = 153° 1' - 115° 50' - 37° 11'

Since we only want to find c, the simplest way to solve this triangle is
to divide it into two right-angled triangles by
drawing a great circle arc B D to cut C A at
right angles.

Then we have from the right-angled triangle
BDC

tan C D = cos C tan a.

tan a = tan 58° 3',
cosC = cos 37° 11',

tan C D, . . .

.-. CD = 51° 56' 47",
and cos c = cos A D . cos B D

10-2050545
9-9012980

10-1063525

Fig. 9«

cos a — cos 58° 3',

cos (b - C D) = cos 10° 35' 13",

cos CD = cos 51° 56' 47",

cos c, .
c = 32° 26' 49".

9-7236026
9-9925435

9-7161461
9-7898616

9-9262845

The circular measure of this angle is -5663.

.-. The distance required = -5663 X 3,960 = 2,242-5 miles.

The more usual method of solving the triangle A B C, having given the

THE EARTH. 29

two sides a, b, and the included angle C, would be to first find the angles
A and B by means of the formulae

cos %(a + 6)

tan £ (A — B) = S~? (ff ~ b) cot
sin f (a + b)

and then find c from the formula

sin C . sin a

sin A

It' this method is adopted to find c, it must be remembered that when
sin c is found there are always two possible solutions, since the sine of an
angle — the sine of its supplement. Some care is, therefore, necessary
in selecting the appropriate value from the two values determined by the
tables.

EXAMPLES FOE SOLUTION.

In all of these examples the earth is to be taken as a sphere of radius
3,960 miles.

1. Find the shortest distance measured along the earth's surface between
Mount Gambier (Longitude 140° 45' E., Latitude 37° 50' S.) and Palmerston
.(Longitude 130° 50' E., Latitude 12° 28' S.).

Ana. 1,856-8 miles.

2. Find the shortest distance measured along the earth's surface between
Baltimore (Lat. 39° 17' N., Long. 76° 37' AY.) and Cape Town (Lat. 33°
.56' S., Long. 18° 26' E.).

Ana. 7,893 miles.

3. How far would a place be due South from the equator if the altitude
of the S. celestial pole was exactly 20° ?

Ans. 1,382-3 miles.

4. Two places are in S. latitude 30°, one longitude 115° E., and the other
35° E. Find the difference in the paths of the two ships sailing from one
port to the other, one along the parallel of latitude and the other along
the arc of the great circle joining the places.

Ans. 1,127 miles.

5. What is the declination of a star that passes through the zenith at a
place in latitude 35° N. ?

Ans. 35° North.

6. A ship sails along the great circle joining two places, each of latitude
45° N., the difference between their longitudes being 2 a. Show that the
highest latitude I reached during the passage is given by the formula

cot / = cos a.

ASTRONOMY FOR SURVEYORS.

7. A ship from latitude 8° 25' N. sails south for 600 miles. What latitude
is she in ?

Ant. lc 35'S.

8. At a place in latitude / North, a star with decimation d rises 60° E,
of North. Show that cos I = 2 sin d.

The Figure of the Earth.— If, as in Fig. 10, F and G
are two points on the same meridian, their difference of
latitude will be measured by the angle FOG. If we
know this angle, and also the length of the arc F G, we

Fig. 10.

shall then be able to calculate the length of the earth's
radius F 0. The difference of latitude between F and G
may be determined by astronomical observation, meas-
uring the altitude of the celestial pole at each place.
The length of the arc F G may be either directly measured
or it may be computed by means of a triangulation survey
from a measured base . line on some suitable adjacent
part of the earth's surface. Determinations of the radius
of the earth on these simple principles were made by the
Greeks 2,000 years ago.

THE EARTH. 31

If the earth were a true sphere, measurements of the
radius of the earth made in this way at different parts
of its surface would be all the same. But when it became
possible to make the necessary observations with suffi-
cient precision it was found that such was not the case.
When Newton discovered and investigated the results
of the law of gravitation in the seventeenth century, he
proved that one consequence was that if the earth is a
plastic body, revolving on an axis and acted on by its
own attraction, it must take the form of a slightly flat-
tened sphere with its polar diameter less than its equatorial
diameter. Measurements of two arcs made by the Cassinis
in France seemed, on the other hand, to indicate that
the length of a degree of latitude decreased towards the
north, which would imply that the shape of the earth
was such that its polar diameter was greater than its
equatorial diameter, contrary to Newton's gravitational
theory. The French Academy equipped two expeditions
in order to settle the problem. One of these measured
an arc in the equatorial regions of Peru (1735-1741), and
the other an arc in the polar regions of Lapland (1736-
1737). The results showed that a degree of latitude was
longer in the polar regions than in parts near the equator,
and corroborated Newton's theory. Since then many
arcs have been measured in different parts of the world,
and the observations have conclusively established the
fact that the shape of the earth is not a true sphere,
but is very approximately an oblate spheroid, the figure
formed by revolving an ellipse about its minor axis.

The shape of the earth is thus like that of a sphere
slightly flattened at the poles. The amount of flattening
is not, however, very great. The length of the earth's
polar axis may be taken as 7,900 miles, and its equatorial
diameter as 7,927 miles. Thus if a model were made 20
feet in diameter, the polar diameter would be shorter than
the equatorial by a trifle over three-quarters of an inch.

ASTRONOMY FOR SURVEYORS.

More exactly still, it is found that the change in the
length of a degree of latitude which takes place as we
proceed along a meridian is not the same along all meri-
dians. It seems that the equatorial section of the earth
is not exactly circular, but is very slightly elliptical.
The exact shape would thus appear to be more nearly
an ellipsoid. For practical purposes, however, all com-
putations in geodetic work are based upon the assumption
that the figure of the earth is an oblate spheroid.

Geographical and Geocentric Latitude. — If in Fig. 11 P
represents some point on the meridian N Q S, N and

8 being the North and South Poles, then, making allowance
for the fact that the section N Q S E is not a circle but
an ellipse, the direction of the horizontal at P will not
be at right angles to P O, O being the earth's centre,
but will be in the direction of the tangent to the ellipse
at P. This is the direction taken by the surface of still
water at that point. .Consequently the direction of the
vertical there is not O P but G P, where G P is the normal
at P — that is to say, it is at right angles to the tangent.
Thus, if we measure the latitude of P by astronomical

THE EARTH. 33

methods, observing the altitude of the celestial pole
above the horizon at P, we shall measure the angle P G Q
and not the angle P O Q. The angle P G Q thus measures
what is called the geographical or geodetic latitude. This
is the ordinary latitude that is used for astronomical
and geodetic purposes.

It is clear, however, that the value of the angle P O Q>
if it can be readily determined, might be equally well
used in order to fix the position of P on the meridian.
This angle measures what is termed the geocentric latitude.
The difference between the geocentric and geographical
latitude of a place is never very great. There is no
difference at all, either at the poles or at the Equator,
and the maximum difference is in latitude 45°, where it
amounts to about 11" 44" of arc. The geocentric latitude
cannot be directly observed. It is computed from the
geodetic latitude by the formula : —

When speaking of latitude in this book, it will always
be the geodetic latitude that is meant unless otherwise
specified.

EXAMPLES.

1. At a place in Lat. 42° S. a line is run from a point A on a bearing
of 220° for a distance of 2,400 chains to a point B.

Assuming the earth a sphere of 3,957 miles radius, find the bearing from
Bto A.

Ans. 40° 15' 12".

2. Given that latitude of London is 51° 32' N., latitude of Jerusalem
32° 44' N., bearing of Jerusalem from London, 1 10° 04'. Find the longitude
of Jerusalem, its distance from London, and the bearing of London from

Jerusalem.

Ans. Longitude, 37° 25' 12" E.
Distance, 2,278 miles.
Bearing of London from
Jerusalem, 316° 00' 16".

34 ASTRONOMY FOR SURVEYORS.

3. The latitude of a Trig. Station A is 33° 51' S., and its longitude is
151° 12' 42" E. The bearing and distance to another Trig. Station B is
284° 08' 44", 105,600 feet.

Compute the latitude and longitude of B, and the bearing of B to A,
on the assumption that the earth is a sphere with radius 20,890,790 feet.

Ans. Longitude, 151° 28' 48" E.
Bearing, 104° 56' 20".

4. Find the great-circle distance in English statute miles from Wellington.
N.Z., to Panama, treating the earth as a sphere, and one degree as equal
to 69 2V statute miles.

Wellington, . . Lat. 41° 17' S., Long. 174° 47' E.
Panama, . . Lat. 9° 00' N., Long. 70° 31' W.

Ans. 4,528-6 miles.

5. Two places are each in latitude 50° N., and their difference of longitude
is 47° 36'. Find their distance apart.

Ans. 2,090 miles.

CHAPTER IV.

THE SUN.

The Sun's Apparent Motion among the Stars. — Like the fixed
stars, the sun shares in the apparent general daily rota-
tion of the heavens, but unlike them it does not always
maintain the same position relative to other objects on
the celestial sphere. In addition to its daily circling of
the sky, it appears to gradually shift its position with
respect to the stars. Neither its declination nor its right
ascension remain^constant . Very little consideration will
show that its declination must alter during the year,
for, if it did not, the sun would always describe the same
circle in the heavens. If this were the case, then, like
the fixed stars, it would always rise and set at the same
points on the horizon, and it would always attain the same
altitude when on the meridian. Since it does not do this,
it is clear that the declination of the sun must change
during the year. That the sun has also a movement in
right ascension among the stars is not quite so obvious,
but the fact may be readily inferred if we watch the stars
that are visible in the East on succeeding mornings just
before sunrise or in the West just after sunset. Stars
in the East that rise just before the sun, so that in a very
short time after rising they are masked by the sun's rays,
will pn each succeeding morning be seen for a longer
time. Similarly stars in the West, setting just after the
sun, will be visible for shorter and shorter periods as we
watch them on successive evenings until finally they are
lost altogether in the strong sunlight, other stars further
East taking their places. Hence we infer that the sun has

36 ASTRONOMY FOR SURVEYORS.

a progressive movement among the stars from West to
East.

The problem of determining the sun's place on the
celestial sphere with regard to the fixed stars was a difficult
one to early astronomers, because as soon as the sun
becomes visible its strong light prevents the stars from
being observed at the same time. Some used the moon,
and Tycho Brahe used the bright planet Venus in order
to get the connection, observing the relative position
of the sun and moon or of the sun and Venus when both
were visible, and afterwards measuring the position of
the moon or Venus with regard to the stars when the
sun had set. But as both the moon and Venus also move
amongst the stars, the movement that had taken place
in the interval had to be allowed for, and the method
was thus not particularly simple. The sun's position is
nowadays determined by much more accurate methods.

The Earth's Orbit round the Sun. — All of these move-
ments of the sun are apparent only and not real. Just
as its apparent daily rotation in the heavens is due to
the rotation of the earth on its axis, so the sun's apparent
movements in right ascension and declination are really
due to the fact that the earth moves in a great orbit
round the sun once a year.

Actually the earth moves round the sun in a path that
is very nearly a huge circle with a radius of about 96
millions of miles. More accurately, the path is described
as an ellipse, one focus of the ellipse being occupied by
the sun. The curve traced out by the centre of the earth
lies in a fixed plane that passes through the centre of the
sun. The earth traces out its complete orbit once a year,
and all the time it is spinning on its own axis once a day,
the direction of the spin on its axis being the same as
that in which it moves round the sun. The earth's axis
is not at right angles to the plane of its orbit, but it makes
with the plane a fixed invariable angle of 66° 32J'. That

THE SUN.

the direction of the earth's axis is constant we know
from the fact that the position of the celestial pole amongst
the fixed stars shows no appreciable shift throughout
the year. Thus, as is illustrated in Fig. 12, the earth
moves round the sun, spinning on its axis, which is inclined
to the plane of the orbit, and the axis always remains
parallel to itself, pointing ever in the same direction
amongst the fixed stars, whose distances, it must be
remembered, are practically infinitely great even in com-

parison with the immense distance of the earth from
the sun.

When the earth is in the position marked 1, the sun will
be shining directly overhead in a place such as a North of
the equator. If e is a point on the earth's equator on
the same meridian of longitude as a, O being the earth's
centre, the angle a O e will be the complement of
66° 32J' or 23° 27|' — that is to say, a will be a point on
the Tropic of Cancer. In this position, then, the sun
at mid-day will be vertically overhead at all points on
the Tropic of Cancer. This statement is not quite accurate,

38 ASTRONOMY FOR SURVEYORS.

because the earth does not remain in the one position
in its orbit while it makes a complete revolution on its
axis ; it is moving forward in its orbit all the time, but
as it takes a whole year to go round the sun, its relative
movement is not very great in one day.

As the earth moves from position 1 to position 2, its
axis always remaining parallel to its original direction,
it will be seen that the sun will appear to shine directly
overhead at points successively nearer and nearer to the
equator, until in position 2 the sun's rays fall vertically
at the equator.

Similarly, as the earth moves on to position 3, the sun's
rays will fall vertically at points further and further
south of the equator, until at position 3 the sun will
appear at mid-day to be overhead at a point on the
Tropic of Capricorn. From there on to position 4 the
sun will shine vertically at points successively nearer
to the equator, until at 4 the sun is once more overhead
at the equator.

The earth is in the position marked 1 on June 2 2nd,
hi that marked 2 on September 22nd, at 3 on December
22nd, and at 4 on March 21st.

Thus, if we consider the appearance of the sun to an
observer at some point P to the south of the Tropic of
Capricorn, on June 22nd the sun will appear to be further
from the zenith and lower down in the sky than at any
other period of the year. On December 22nd, when the
earth is in position 3, the sun at mid-day will be nearer
the zenith than at any other time of the year.

The orbit of the earth being an ellipse, its distance
from the sun is not constant. It is furthest from the
sun in the position 1, and nearest to the sun in the
position 3.

The Equinoxes. — On March 21st and September 22nd,
the sun, being vertically overhead at the equator, will
appear to an observer at any part of the earth to be in

THE SUN. 39

the celestial equator. Now, we have seen that when
any heavenly body is in the celestial equator its path
is bisected by the horizon, so that the time during which
it can be seen in the sky is equal to the time during which
it is invisible. Thus, when the earth is in either of these
positions the days and nights are of equal length all over
the world. These points are consequently called the
Equinoxes.

Motion in Right Ascension and Declination. — It thus appears
that on March 21st and September 22nd the sun's
declination is zero, as it lies on the Celestial Equator.
From March 21st to September 22nd it will appear in
the sky to the North of the equator, so that its declina-
tion will be north with a maximum value of 23° 27|' on
June 22nd. From September 22nd to March 21st its
declination will be south with a similar maximum value
on December 22nd.

It is also evident that the sun's right ascension changes
throughout the year, because as the earth revolves round
it the apparent position of the sun among the fixed stars
must obviously change. The stars that would be seen
by an observer on the earth when in position 1, looking
in the direction of the sun, would be seen by an observer
at 3 when looking in the direction opposite to that of the
sun. Clearly, in the course of the year the sun will trace
out a complete circle among the fixed stars.

The declination and right ascension of the sun are given
in the Nautical Almanac for Greenwich noon on every
day of the year. The values at intermediate instants
may be found by interpolation. Illustrations of such
calculations are given in Chapter VIII. when dealing with
sun observations.

The Sun's Semi-Diameter. — The disc of the sun sub-
tends at the eye of an observer an angle of about half a
degree. By accurately measuring the angle subtended
by diameters taken in different directions, we find that

40 ASTRONOMY FOR SURVEYORS.

these are all equal, so that the disc is circular in form.
In order to mark the position that the sun occupies on
the celestial sphere at any time, we require to determine
the position of the centre of the circular disc. But there
is no mark at the centre that we can recognise, and so
in practice we must observe a point on the edge of the
sun and then make an allowance for the distance of this
point from the sun's centre.

From what we have just seen of the nature of the
earth's motion round the sun, it is clear that the sun
is not at all times of the year at the same distance from
us, and consequently we should not expect its diameter
to remain constant. As the earth completes its orbit
round the sun in a year and then goes over the same
path again, we might anticipate that the variations in
the value of the sun's apparent diameter would follow
a yearly cycle. This is found to be the case, a slow de-
crease taking place from the 31st of December to the
of July, and a slow increase during the second half

the year.

As the semi-diameter is frequently required in reducing
sun observations, the values are chronicled for every day
in the year in the Nautical Almanac (p. 11 of each month).
In the almanac for 1914 the maximum value of the semi-
diameter is given on January 3rd as 16' 17-55", and the
minimum on July 3rd as 15' 45 -38".

To Plot the Position of the Sun's Centre on the Celestial
Sphere. — Supposing that we know the direction of the
true North and South, and also the latitude of the place
of observation, we may readily measure the declination
of the sun at mid-day. With a telescope pointed in the
direction of the meridian we may observe the altitude
of the sun's upper or. lower edges (limbs, as they are
usually called) at the moment when it crosses the meridian.
Making due allowance for the sun's semi-diameter, we
shall thus obtain the meridian altitude of the sun's centre.

THE SUN. 41

Thus, as in Fig. 13, if P represents the Pole, Z the zenith,
we measure either Sx N or S2 S, according as the sun is
in a position such as Si or as S2. Now, we have previously
shown that the altitude of the celestial pole, P N, is
equal to the latitude of the place. Thus, if the sun is
situated as at S15 on the same side of the zenith as the
pole, the difference between the observed altitude Sx N
and the latitude P N gives the sun's polar distance P S,.
If the sun is at S2, on the opposite side of thejzenith
to the pole, then the arc S2 N is equal to 1802-^the observed
altitude S S2. The difference between S2 N and the
latitude P N gives the sun's polar distance as before.
The declination of the sun is the complement of its
polar distance.

Fui. 13. /

Having measured the declination of the sun in this
way, in order to fix its position on the celestial sphere,
it only remains to determine the difference between its
right ascension and that of some star whose co-ordinates
are known. But we have seen that the difference of right
ascension of any two stars is measured by the interval
in time between their transits across the meridian, as
given by the sidereal clock. If, with the sidereal clock,
the times be measured when the first and second limbs
of the sun cross the meridian, the mean of the two times
will give the instant when the centre crosses the meridian.
If, therefore, the time of passage across the meridian of
some selected known star is also observed, the interval

42 ASTRONOMY FOR SURVEYORS.

between the two times, reduced to degrees, will give
the difference between the right ascension of the sun
and the star.

These observations give us the elements necessary to
plot the position of the sun.

The Sun's Apparent Annual Path on the Celestial Sphere.
— In Fig. 14, let A represent the position of the selected
fixed reference star as plotted on a globe representing
the celestial sphere, P being the Pole, Q R the great
circle of the equator, and S N the horizon. Then, if we

Fig. 14.

set out the angle A P B equal to the observed difference
of right ascension and measure off the arc P B equal
to the observed polar distance of the sun, the point B
will represent the position of the sun's centre on the star
globe.

When observations similar to those just described are
made day after day, and the corresponding positions
of the sun plotted on -the globe, those positions are all
found to lie on a great circle, which cuts the equator at
two opposite points <v» and jfi in the figure, and is
inclined to it at an angle of about 23° 27'.

The great circle, the plane of which contains the sun's
yearly path, is called the ecliptic'l&nd the angle this makes
with the equator is spoken of/ as the obliquity of the ecliptic.

Its points of intersection with the equator are called
the equinoctial points, one (<¥») is known as the First Point
of Aries, and the other (^) as the First Point of Libra.

The sun is at the first of these points on about the
21st of March (the vernal equinox), and at the second on
the 23rd of September (the autumnal equinox), its decli-
nation being then 0° and its polar distance 90°.

As we have already seen, <¥ is the point selected on the
equator as that from which right ascensions are measured,
so that the right ascension of ^ is 0° and that of -°- 180°.

At the two points on the ecliptic whose right ascensions
are respectively^0 and 270°, the sun will have its greatest
declination north and south of the equator. These are
known as the Solstitial Points. The sun reaches them
on or about the 22nd of June and the 22nd of December.
On June 22nd the sun has its greatest declination of about
23° 27' north of the equator, and on December 22nd its
greatest declination south.

EXAMPLES.

1. Determine the meridian altitude of the sun at a place in latitude
30°, (a) at the equinoxes, (6) during the summer solstice.

Ans. 60° and 83° 27'.

2. Find the latitude of the place where the greatest altitude of the sun
in midsummer is 60°. /

Ans. 53° 27'.

3. At a place in lat. 80° N., on a certain day the sun at mid-day just-
appears above the horizon. Find the sun's declination. 'Find also the
altitude of the sun at mid-day when its declination is 20° N.

Ans. 10° S. and 30°.

CHAPTER V.

TIME.

Sidereal Time. — To measure time we require some form
of perfectly uniform motion, and the most perfect motion
of this kind in the heavens is provided by the apparent
revolutions of the fixed stars. The earth turns on its
axis with absolutely regular speed and, as the stars are
so distant that the movement of the earth in its orbit
round the sun produces no apparent effect upon their
relative positions, the consequence is that the stars
complete a revolution round the celestial pole at a per-
fectly regular rate in a fixed and constant time. To the
astronomer, then, this presents the simplest way of
measuring time. The period of a complete revolution
of the stars round the pole is known as the sidereal day,
and time measured in this way is termed sidereal time.

Apparent Solar Time. — Convenient as the above method
of measuring time is to the astronomer, it is obviously
unsuited to ordinary purposes of life. It is the day as
determined by the sun that controls our habits and rules
our lives. The apparent solar day, or period of time
between successive transits of the sun across the meridian,
is, however, variable in length, and it is impossible to
regulate a clock so that it shall indicate exactly 12 o'clock
just when the sun is in the meridian. The reason of this
may be seen from Fig. 15, which shows in an exaggerated
way the movement of the earth in its orbital revolution
round the sun. Suppose that, when the earth is in the
position marked 1, the sun is directly overhead to an
observer at A, and that, if it could be seen, the star F

TIME. 45

would appear in the same direction. As the earth revolves
on its axis it also travels forward in its orbit, so that
at the end of a sidereal day it is in the position marked 2!
If the observer has been carried round to the point B, so
that the same star F appears vertically overhead, the
star being at practically an infinite distance, B F will be
parallel to A F. The interval between these two positions
marks a sidereal day. But to bring the sun overhead,
to the same observer, he must wait till he is carried round
the extra distance B C. The solar day then will be longer

•®&.

Fig. 15.

than the sidereal day by the length of time required to-
traverse this extra distance. Whilst the sidereal day is
the time taken by the earth to make a complete revolu-
tion on its axis, the apparent solar day is the time taken
to make a little more than a revolution.

Now, the earth does not move in a circular but in an
elliptic orbit round the sun, so that sometimes it is nearer
to the sun than at others. When it is nearer to the sun
it is a deduction from the law of gravitation that it must

46 ASTRONOMY FOR SURVEYORS.

travel faster in its path than when it is further away.
The result is that the extra little bit, B C, through which
fhe earth has to turn in the interval of time that has to
be added on to the sidereal day to give the apparent solar
day, is not always the same, and the apparent solar day
is thus not of constant length.

We have seen that the right ascensions of the fixed
stars are practically constant. But if a celestial body
were to move in right ascension its period of revolution
about the pole would still be constant, although not
the same as that of the stars, provided the movement
was a uniform one. The difficulty with the sun as a time-
keeper is that its motion in right ascension is variable.

Mean Time. — The right ascension of the real sun
changes by 360° in the course of a year, but the rate of
change is not always the same. We might conceive of
an imaginary body travelling with the sun, so that its
right ascension changes by the same amount in the course
of the whole year, but having its motion in right ascension
perfectly uniform. Such an imaginary sun would form
a, perfect time-keeper, we could regulate our clocks to
mark noon when it should be on the meridian, and it
would have the great practical advantage that the time
so indicated would never be very far different from that
of the actual sun. This imaginary sun is termed the
mean sun, and the time indicated by it is called mean
solar time. The mean sun is pictured as moving along
the equator with uniform speed, so that its motion is
the average of that of the actual sun in right ascension.
A mean solar day is the interval between two successive
transits of the mean sun across the meridian.

The Three Systems of Time Measurements. — There are
thus three kinds of time -to be considered.

1. Sidereal, as determined by the revolution of the stars.

2. Apparent solar, as measured by the actual sun or a

sun dial.

TIME. 47

3. Mean solar, which is the ordinary time kept by our
clocks.

The hour angle (Chap. II.) of the real sun gives the
apparent time or time indicated by a sun dial, and the
hour angle of the mean sun gives the mean time at that
instant.

Mean noon is the instant when the mean sun is on the
meridian. The mean time at any other instant is measured
by the hour angle of the mean sun reckoned westward
from 0 hr. to 24 hrs. Thus the astronomical mean day
is usually divided into 24 hours instead of the two divisions
of 12 hours each in common use for civil purposes. As
the astronomical day starts at noon, both methods will
agree in the afternoon of each day, but not in the morning.
Thus, July 29th, 10 p.m., would be the same in both
the civil and astronomical methods of reckoning, but
July 29th, 10 a.m., Civil time, would be equivalent to
July 28th, 22 hrs., astronomical time.

Equation of Time. — The difference between the mean
and the apparent solar time is known as The Equation of
Time. It is counted positive when the mean time exceeds
the apparent time, and negative when the apparent
time is greater than the mean. It is thus always the
amount that must be added to the apparent to obtain
the mean time. Thus we have —

Mean Time = apparent Time + Equation of Time.
or Clock Time = sun dial Time + Equation of Time.

When the actual sun is on the meridian, the sun dial
will indicate 0 hr. or noon. Hence —

Equation of Time = mean Time of apparent noon.

The equation of time is thus positive if the sun is " after
the clock/' or the true sun transits after the mean sun.
Its values at both mean and apparent noon at Greenwich
are tabulated in the Nautical Almanac for every day in
the year.

48 ASTRONOMY FOR SURVEYORS.

The equation of time vanishes four times a year, on
or about April 15th, June 15th, September 1st, and
December 24th. From December 24th till April 15th it
is positive, with a maximum value of about 14 min.
26 sec. on February llth. From April 15th to June 15th
it is negative, having its greatest value of about 3 min.
48 sec. on May 15th. From June 15th to September 1st
it is again positive with a maximum value of about
6 min. 19 sec. on July 27th. Between September 1st and
December 24th it is negative once more, attaining its
greatest negative value for the year, about 16 min. 21 sec.
on November 3rd. These dates are approximate only,
as they are not always precisely the same in different
years.

It will be seen on looking at the tabulated values of the
equation of time in the Nautical Almanac, that it is a
continuously varying quantity, its value commonly
changing by several seconds from one day to the next.
The tabulated values are for Greenwich noon, and con-
sequently if we wish to know the equation of time at some
other instant we must find its value by interpolation.
To facilitate this the Nautical Almanac gives the value
of the variation in one hour at each noon.

For example, the equation of time at Greenwich mean
noon on March 21st, 1913, is given as 7 min. 25-89 sec.,
and is diminishing from day to day. The variation in
one hour at noon on March 21st is 0-755 second. If,
then, we require the equation of time at 11 hrs. on March
21st (Greenwich time), all we have to do is to subtract
11 X 0-755 sec. from 7 min. 25-89 sec., giving, as the
equation of time at the required instant, 7 min. 17-59 sec.

If it is desired to make the computation with the
greatest precision, allowance must be made for the fact
that the rate of variation given is the rate at Greenwich
noon, and not the mean rate over the 11 hours. The rate
of variation at noon on the next day, March 22nd, is

TIME. 49

given as 0-760 sec., and, therefore, the rate of variation

Oil

5| hours after noon on March 21st is 0-755+ ~ x 0-005

= 0-756. This would more accurately represent the
mean rate of variation during the 11 hours, and the
required equation of time is, therefore, more accurately,
7 min. 25-89 sec.- 11 x 0-765= 17 min. 17-57 sec.

The more accurate procedure thus only makes a differ-
ence in the second place of decimals of a second, and the
simpler method given at first is good enough for most
purposes.

EXAMPLE. — Find the equation of time at 5 hrs. 30 min. on February 25th,
the equation of time at noon being 13 min. 17-86 sec. and the variation
in one hour 0-395 sec.

Ans. 13 min. 15-69 sec.

Local Mean Time — -The local mean time at any place
is reckoned by counting as 0 hr. the instant when the
mean sun last crossed the meridian of the place. As the
earth rotates uniformly on its axis from West to East,
it follows that the further East a place is situated the sooner
will the sun cross the meridian, and, therefore, the later
will be the local time. All places on the same meridian
of longitude have their noons at the same instant, and,
as the earth turns, one meridian after another is brought
opposite to the sun. Thus, the interval of time between
the local noons at two different places will depend upon
their difference of longitude.

As the earth turns through 360° in 24 hours, it follows
that a difference of 15° of longitude corresponds to a
difference of 1 hour in time, 15' of arc corresponds to a
difference of 1 minute of time, and 15" of arc to a difference
of 1 second of time.

Thus, if we know the longitude and the local time at
one place A, we can readily compute the time at any
other place B whose longitude is given. We have only

50 ASTRONOMY FOR SURVEYORS.

to convert the difference of longitude into time, at the
rate of 15° per hour, and add this to the time at A if B
is to the East, or subtract it if B is to the West from A.

EXAMPLE.— If the longitude of A is 36° 03' 37" E., and the local mean
time is September 5th 1 hr. 31 min. 17 sec., find the time at B in longitude
3° 27' 13" E.

The difference of longitude = 32° 36' 24".

To convert this into time, we simply have to divide by 15, giving us,
as the difference in time between the two places, 2 hrs. 10 min. 25-6 sec.

As B is to the West from A, this has to be subtracted from 1 hr. 31 min.
17 sec., giving us as the time at B, September 4th, 23 hrs. 20 min. 51-4 sec.

Should one longitude be East from Greenwich and the
other West, we must add them, instead of subtracting,
in order to get the angle between the meridians.

EXAMPLE. — A ship sails from London on January 2nd at 1 p.m., and
arrives in Melbourne (longitude 145° E.) at 6 p.m. on February 8th. Find
the time occupied by the voyage.

Ans. 36 days 19 hrs. 20 min.

Local Sidereal Time. — The local sidereal time at any
place is reckoned by counting as 0 hr. the instant when
the First Point of Aries last crossed the meridian of the
place. Therefore, in precisely the same way, if we know
the longitudes of two places A and B and the local sidereal
time at A, we can compute the corresponding sidereal
time at B. For the earth turns on its axis through 360°
relative to the fixed stars in 24 sidereal hours, and, there-
fore, a difference of longitude of 15° corresponds to a
difference of 1 hr. in the sidereal times. The method
to be used for finding the sidereal time at B is thus exactly
the same as that just illustrated.

EXAMPLE. — If the sidereal time at A, long. 35° E is 12 hrs. 30 min., find
the sidereal time at the same instant at B, long. 27° \\ .

Ami, 8 hrs. -2-2 min.

Apparent Solar Times at the Same Instant at Places in
Different Longitudes.— The equation of time or difference
fbetween apparent and mean times is the same all over

TIME.

the world at the same instant. Consequently the difference
between the apparent solar times at two places A and B
is precisely the same as the difference between the local
mean times. The same method again then can be used
to determine the apparent time at B, having given the
apparent time at A.

EXAMPLE. — If the apparent solar time at A, long. 45° W. is 1 hr: 30 min.»
and the equation of time is 6 min. 10 sec., to be added to apparent time,
find the corresponding mean time at B in longitude 10° W.

Ans. 3 hrs. 56 min. 10 sec.

Standard Time. — To avoid the confusion arising from
the use of different local times in each town, most countries
now adopt the system of using the time on a particular meri-
dian through the country that lies an even number of hours
from Greenwich. The following table shows the standard
times adopted by the principal countries of the world :—

Longitude of Standard Meridan.

Countries.

In Degrees.

In Time.

Hrs. Min.

172£° E.

11 30 E.

New Zealand.

150° E.

10 0 E. Victoria, New South Wales,

Queensland, Tasmania.

142£° E.

9 30 E. South Australia.

135° E.

9 0 E. Japan, Corea.

120° E.

8 0 E. Western Australia.

82£° E.

5 30 E. India.

30° E.

2 0 E.

East Europe, South Africa, Egypt.

15° E.

1 0 E.

Germany, Austria, Denmark,

Sweden, Norway, Switzerland,

Italy, Western Turkey.

0°

0 0

Great Britain, Belgium. Spain.

60° W.

4 0 W.

Atlantic Provinces of Canada.

75° W.

5 0 W.

Quebec, Eastern Zone of the

United States, Peru.

90° W.

6 0 W.

Central Zones of Canada and

U.S.A.

105° W.

7 0 W.

Mountain Zones of Canada and

U.S.A.

120° W.

8 0 W.

British Columbia and the Pacific

Zone of U.S.A.

52 ASTRONOMY FOR SURVEYORS.

To Change Standard Time to Local Mean Time. — This
problem has really been already discussed, for the differ-
ence between standard time and local mean time at any
place is that due to the difference of longitude between
the given place and the standard time meridian used.
For places East of the standard meridian local mean time
is later than standard time, and for places to the West
the local time is earlier.

EXAMPLES.

The standard time meridian in South Australia being 142° 30' E., find
the local mean time at Adelaide (longitude 138° 35' E.) when the standard
time is 8 hrs. 25 min. 10 sec.

Ans. 8 hrs. 9 min. 30 sec.

In New York State the standard time meridian is 75° W. If the local
mean time is 10 hrs. 17 min. 18 sec. at a place in the State, the longitude
of which is 73° 58' W., find the standard time.

Ans. 10 hrs. 13 min. 10 sec.

To Reduce a Given Interval of Mean Time to Sidereal Time
and vice versa. — It will be seen from the consideration
of Fig. 15 that in the course of its complete orbital
revolution round the sun the earth will make exactly
>one turn less with respect to the sun than it does with
respect to the fixed stars. There are approximately
365J mean solar days in the year, and, therefore, in the
same period there are 366 J sidereal days. More exactly,
according to Bessel, the year contains 365-24222 solar
days, and hence 365-24222 solar days^- 366-24222 sidereal
days.

Therefore, if m be the measure of any interval in mean
time and s the corresponding measure in sidereal time,

ra_ 365-24222
*T~ 366 -24222*

Thus, if m be given, s can be found, or vice versa.
Tables to facilitate the reduction are Driven in the

TIME.

Nautical Almanac, and less elaborate ones in Chambers'
Mathematical Tables.

When tables are not used, the simplest way to make
the computation is as follows : —

To convert an interval of mean solar time to sidereal
time, add 9-8565 seconds for each mean solar hour.
Dividing by 60, this gives us -1642 second to be added
for each minute and -0027 second for each second of
mean time.

Thus, to convert an interval of 6 hrs. 33 min. 17 sec.
of solar time into the equivalent interval of sidereal time,
we have —

6x 9-8565= 59-139
33 x -1642= 5-418
17^x -0027- -046

64-603 seconds = 1 min. 4-6 sec.

The addition of this to the given solar time gives us
6 hrs. 34 min. 21-6 sec. as the equivalent sidereal
interval.

To convert an interval of sidereal time to the equivalent
interval of mean solar time, subtract 9-8296 seconds for
each sidereal hour. Dividing by 60 we get -1638 second
to be subtracted for each sidereal minute, or -0027 second
for each second.

Thus, to find the interval of solar time equivalent to
an interval of 6 hrs. 33 min. 17 sec. of sidereal time, we
have —

6x 9-8296= 58-978
33 x -1638- 5-405
17 x -0027= -046

64 -42 9 seconds = 1 min. 4-43 sec.
Subtracting this from the given interval of sidereal

54 ASTRONOMY FOR SURVEYORS.

time gives 6 hrs. 32 min. 12-57 sec. as the equivalent mean
time interval.

Given the Sidereal Time at Mean Noon at Greenwich on any
given Date to find the Local Sidereal Time at Local Mean Noon at
any other Place on the Same Date.

On page 11 for each month in the Nautical Almanac
the Greenwich sidereal times are tabulated for Greenwich
mean noon on each day. From these it is necessary,
in most work in which the time has to be brought into the
calculations, that we should be able to deduce the local
sidereal time at local mean noon on the corresponding
day at the place of observation.

In the succeeding pages it will be convenient to use the
following abbreviations : —

G.M.T. to denote Greenwich mean Time.
G.S.T. „ Greenwich sidereal Time.

G.M.N. „ Greenwich mean noon.
L.M.T. „ Local mean Time.

L.S.T. ., Local sidereal Time.

L.M.N. ,, Local mean noon.

From what we have already done, it will be evident
that if we have two clocks, one set to keep sidereal time
and the other to keep mean time, the sidereal clock will
complete its day in a shorter period than the other, and
consequently will be continually gaining. According to
the last article, it will gain at the rate of 9-8565 seconds
for each solar hour.

Now, at a place in West Longitude, noon occurs a certain
number of hours after noon at Greenwich, the interval
depending upon the longitude. But the tabulated sidereal
time at Greenwich noon is the difference between the
readings of the sidereal and mean time clocks at that
instant. Consequently, by the time it becomes noon
at the place in question, the sidereal time will have gained
still further on the mean time clock at the rate of 9-8565

TIME 55

seconds for each hour of longitude. Thus the L.S.T. at
L.M.N. will be greater than the G.S.T. at G.M.N. by an
amount computed at the rate of 9-8565 seconds for each
hour of West longitude.

Similarly, at a place in East Longitude, noon occurs
before the corresponding noon at Greenwich, and in this
case L.S.T. at L.M.N. will be less than the G.S.T. at
G.M.N. by an amount computed in the same way according
to the longitude.

EXAMPLE.— On October 1st, 1914, the G.S.T. at G.M.N. is given in the
Nautical Almanac as 12 hrs. 37 min. 29-99 sec. Determine the L.S.T. at
L.M.N. (a) at a place in longitude 57° 33' 28" West, (6) at a place in the same
longitude East.

(a) 57° 33' 28" is equivalent to 3 hrs. 50 min. 13-87 sec.

3 x 9-8565 = 29-569

50 x 0-1642 = 8-210

13-87 x -0027 = -037

37-816, say 37-82 sees.

Therefore, for a place in West longitude we must add this on to the 12 hrs.
37 min. 29-99 sec., giving 12 hrs. 38 min. 07-81 sec. as the L.S.T. at L.M.N.

(6) If the place is in East longitude, we must subtract the 37-82 seconds,
giving 12 hrs. 36 min. 52-17 sec. as the L.S.T. at L.M.N. in that case.

EXAMPLE.— On December 1st, 1914, the G.S.T. at G.M.N. is 16 hrs. 37 min.
59-89 sec. Compute (a) the G.S.T. at G.M.N. on December 2nd, (6) the
L.S.T. at a place in longitude 43° 35' West at L.M.N. on December 1st.

Ans. (a) 16 hrs. 41 min. 56-45 sec.
(6) 16 hrs. 38 min. 28-52 sec.

Given the Local Mean Time at any Instant, to Determine the
Local Sidereal Time.

The local mean time gives us the interval measured in
solar hours, minutes, and seconds, that has elapsed since
local noon. We may readily turn this interval into
sidereal hours, and so obtain the number of sidereal
hours, minutes, and seconds that have elapsed since
noon. But in the preceding paragraph we have seen
how the L.S.T. at L.M.N. may be determined on any

56 ASTRONOMY FOR SURVEYORS.

given date at a place in any longitude. Consequently
we have only to add to this the number of sidereal hours,
minutes, and seconds that have since elapsed, to deter-
mine the sidereal time at the instant. We. therefore,
proceed as follows :—

1. From the tabulated G.S.T. of G.M.N. on the date
in question, compute the L.S.T. of L.M.N. by allowing
for difference in longitude.

2. Turn the given L.M.T. into sidereal hours, minutes,
and seconds, and add to the L.S.T. of L.M.N.

EXAMPLE. — Find the sidereal time at Mount Hamilton
(Longitude 121° 38' 43-35" West) on October 2nd, 1913,
the L.M.T. being 9 hrs. 17 min. 32 sec. p.m.

Dividing the longitude by 15, we get the difference in local times between
Mount Hamilton and Greenwich to be 8 hrs. 06 min. 34-89 sec.

The gain of the sidereal over the mean time clock in this interval, at the
rate of 9-8565 seconds per hour, is 1 min. 19-93 sec.

From the Nautical Almanac, we get G.S.T. at G.M.N. on October 2nd,

1913, 12 hrs. 42 min. 23-50 sec.

Add, . 0 hr. 1 min. 19-93 sec.

L.S.T. at L.M.N., . 12 hrs. 43 min. 43-43 sec.

But 9 hrs. 17 min. 32 sec. of mean time,

when turned into sidereal time, . . 9 hrs. 19 min. 03-59 sec.

Therefore, L.S.T. required, . . . 22 hrs. 02 min. 47-02 sec.

EXAMPLE. — Find the sidereal time at Adelaide (longitude
138° 35' 04-5" E.) on October 2nd, 1913, the standard time
being 9 hrs. 17 min. 32 sec. p.m.

The standard time for South Australia is that of the meridian 142£° or
9 hrs. 30 min. E.

Difference in local times between Adelaide and Greenwich = 9 hrs. 14 min.
20-3 sec.

The gain of the sidereal over the mean time clock in this interval at the
rate of 9-8565 seconds per hour is 1 min. 31-06 sec.

G.S.T. at G.M.N. on October 2nd, 1913, 12 hrs. 42 min. 23-50 sec.
Subtract, 0 hr. 1 min. 31-06 sec.

L.S.T. at L.M.N 12 hrs. 40 min. 52-44 sec.

TIME. 57

The difference between local time and standard time is 15 min. 39-7 sec.
Therefore, the local mean time is . . 9 hrs. 01 min. 52-3 sec.

Turning the interval into sidereal time,
we get 9 hrs. 03 min. 21-31 sec.

Therefore, L.S.T. required, . 21 hrs. 44 min. 13-75 sec.

It is to be particularly noticed that the local mean
time must always be reckoned from noon when making
such calculations.

Thus, if the mean time is given as 9 hrs. a.m. on October
2nd, this must be reckoned as 21 hrs. October 1st, or
21 hrs. after noon on October 1st.

Given the Sidereal Time at a Place whose Longitude is known,
to Determine the corresponding Local Mean Time.

If we can find the sidereal time at m^an nnnn1 then
by subtracting this from the given sidereal time we find
the number of sidereal hours, minutes, and seconds that
have elapsed since noon. Turning this interval of time
into mean time will give us the number of mean time
hours, minutes, and seconds since noon — that is to say,
the mean local time required. The rules of procedure
are thus :—

1. From the tabulated G.S.T. of G.M.N. on the date
in question, compute the L.S.T. of L.M.N. by allowing
for difference in longitude.

2. Subtract the L.S.T. of L.M.N. from the given sidereal
time. Turn the difference into mean solar time, and the
result will be the mean time required.

EXAMPLE. — Given that the sidereal time at Mount Hamilton
is 22 hrs. 02 min. 47-02 sec. on October 2nd, 1913, the
longitude of the place being 121° 38' 43-35" West, find the
corresponding local mean time.

As in the first example of the preceding section, we obtain L.S.T. at

L.M.N., 12 hrs. 43 min. 43-43 sec.

Given sidereal time, . . . .22 hrs. 02 min. 47-02 sec.

Difference, ... 9 hrs. 19 min. 03-59 sec.

58 ASTRONOMY FOR SURVEYORS.

Turning this interval into mean solar time, by the aid of the tables,
we get 9 hrs. 17 min. 32 sec. as the L.M.T. required.

EXAMPLE. — Given that the sidereal time at Adelaide
(longitude 138° 35' 04-5" E.) is 21 hrs. 44 min. 13-75 sec.
on October 2nd, 1913, find the corresponding local mean
time.

As in the second example of the preceding section, we obtain L.S.T.

at L.M.N., 12 hrs. 40 min. 52-44 sec.

Given sidereal time 21 hrs. 44 min. 13-75 sec.

Difference, . . . 9 hrs. 03 min. 21-31 sec.
Turning this interval of sidereal time into mean time, we obtain 9 hrs.
01 min. 52-3 sec. as the L.M.T. required.

Alternative Method for Determining the L.S.T., having given
the L.M.T. — In the preceding methods for computing
L.S.T. from L.M.T. or vice versa, it is necessary to first
of all compute the L.S.T. of L.M.N., and then to trans-
form another interval of time from mean to sidereal or
from sidereal to mean. In the methods about to be
described the theory is perhaps a little more complex,
but there is only one transformation of a time interval
necessary, so that the actual computation is a little
shorter.

From the given L.M.T., allowing for the difference of
longitude, we readily compute the corresponding mean
time at Greenwich. This gives us the interval in mean
time that has elapsed since the last Greenwich noon.
Turn this interval into sidereal time, and we get the
number of sidereal hours, minutes, and seconds that
have elapsed since the mean sun was last on the Green-
wich meridian.

But from the Nautical Almanac we get the G.S.T. at
the last G.M.N. Allowing for the difference in longitude,
we can thus obtain the L.S.T. at that instant. And as
we have already computed the interval in sidereal time

TIME. 59

that has since elapsed, we have only to add this on to the
L.S.T. at the preceding G.M.N. in order to get the sidereal
time required.

We thus get the following rules of procedure :—

1. Allowing for the difference of longitude, compute
the mean time at Greenwich at the instant in question,
and turn the interval of mean time so found into sidereal
time.

2. From the Nautical Almanac obtain the G.S.T. at
the previous G.M.N. , and allowing for the difference of
longitude, determine the corresponding L.S.T. at the
same instant.

3. The addition of the results of 1 and 2 gives the
L.S.T. required.

As illustrations, for purposes of comparison, we will
take the same examples as those already worked.

EXAMPLE. — Find the sidereal time at Mount Hamilton
(longitude 121° 38' 43-35" West) on October 2nd, 1913, the
L.M.T. being 9 hrs. 17 min. 32 sec. p.m.

L.M.T. at Mount Hamilton, . . 9 hrs. 17 min. 32 sec.
Difference due to Longitude (W.), . 8 hrs. 06 min. 34-89 sec.

Corresponding G.M.T., . . .17 hrs. 24 min. 06-89 sec.

Turned into sidereal time, this is equivalent to 17 hrs. 26 min. 58-41 sec.
From the Nautical Almanac we get G.S.T. at G.M.N. on October 2nd,

1913, 12 hrs. 42 min. 23-50 sec.

Difference due to longitude 8 hrs. 06 min. 34-89 sec.

.-. L.S.T. at G.M.N., ... 4 hrs. 35 min. 48-61 sec.

Interval of sidereal time since elapsed . 17 hrs. 26 min. 58-41 sec.

.-. L.S.T. required, .... 22 hrs. 02 min. 47-02 sec.

60 ASTRONOMY FOR SURVEYORS.

EXAMPLE. — Find the sidereal time at Adelaide (longitude
138° 35' 04-5" E.) on October 2nd, 1913, the standard
time being 9 hrs. 17 mm. 32 sec. p.m.

The standard time for South Australia is that of the meridian 142£°
or 9 hrs. 30 min. E.

Standard time at instant, . . . 9 hrs. 17 min. 32 sec.

Subtract difference due to longitude, . 9 hrs. 30 min. 0 sec.

Corresponding G.M.T. on October 1st, . 23 hrs. 47 min. 32 sec.

Turning the interval into sidereal time we get 23 hrs. 51 min. 26-5 sec.
From the Nautical Almanac we find

G.S.T. at G.M.N. on October 1st, . . 12 hrs. 38 min. 26-95 see.
Difference due to longitude of Adelaide, 9 hrs. 14 min. 20-3 sec.

.-. L.S.T. at G.M.N. on October 1st, . 21 hrs. 52 min. 47-25 sec.
Interval of sidereal time since elapsed, . 23 hrs. 51 min. 26-5 sec.

.-. L.S.T. required, .... 21 hrs. 44 min. 13-75 sec.

Alternative Method for Determining the L.M.T., having given
the L.S.T. — Knowing the longitude of the place, we
can compute the sidereal time at Greenwich at the same
instant. From the Nautical Almanac, as before, we get
the G.S.T. at the previous G.M.N. Subtracting these two
results gives us the interval in sidereal time that has
elapsed since Greenwich noon.

If we turn this interval into mean solar time, we, there-
fore, get the interval of mean time that has elapsed since
G.M.N. But the L.M.T. corresponding to G.M.N. is readily
determined by allowing for the difference of longitude.
Adding to this, therefore, the interval of mean time that
has since elapsed, we obtain the L.M.T. required.

The principal difficulty arises in places with East
longitude, where it may happen that the instant under
consideration really precedes noon on the same day at
Greenwich. This cannot happen with places having
West longitude. If this is the case, it will be at once

TIME. 61

noticed from the fact that the sidereal time at Greenwich
mean noon on the day in question, as found from the
Nautical Almanac, will be less than the computed Green-
wich sidereal time at the instant.

We thus get the following rules for determining the
L.M.T., having given the L.S.T. :—

1. Allowing for the difference of longitude, compute the
G.S.T. at the instant in question.

2. From the Nautical Almanac find the G.S.T.
at the previous G.M.N. and then by subtraction
the number of sidereal hours that have elapsed
since. Turn this interval of sidereal time into mean
time.

3. Add this interval of mean time on to the L.M.T.
corresponding to G.M.N. , and the result is the L.M.T.
required.

EXAMPLE. — Given that the sidereal time at Mount
Hamilton is 22 hrs. 02 min. 47-02 sec. on October 2nd>
1913, the longitude of the place being 121° 38' 43-35" West,
find the corresponding L.M.T.

L.S.T. at Mount Hamilton, . . 22 hrs. 02 min. 47-02 sec.

Difference due to longitude (W.), . 8 hrs. 06 min. 34-89 sec

Corresponding G.S.T., . . . 30 hrs. 09 min. 21-91 sec
G.S.T. at G.M.N., October 2nd, 1913, . 12 hrs. 42 min. 23-50 sec.

Interval of sidereal time since G.M.N., . 17 hrs. 26 min. 58-41 sec.

Equivalent interval of mean time, . 17 hrs. 24 min. 06-89 sec.
L.M.T. corresponding to G.M.N., October
2nd = October 1st, . 15 hrs. 53 min. 25-11 sec.

.-. L.M.T. required = October 2nd, . 9 hrs. 17 min. 32 sec.

62 ASTRONOMY FOR SURVEYORS.

EXAMPLE. — Given that the sidereal time at Adelaide
(longitude 138° 35' 04-5" E.) is 21 hrs. 44 min. 13-75 sec.
on October 2nd, 1913, find the corresponding L.M.T.

L.S.T. at Adelaide, . . . 21 hrs. 44 min. 13-75 sec.

Difference due to E. longitude, . . 9 hrs. 14 min. 20-30 sec.

Corresponding G.S.T., . 12 hrs. 29 min. 53-45 sec.

G.S.T. at G.M.N., October 2nd, 1913, . 12 hrs. 42 min. 23-50 sec.

Instant precedes G.M.N. by 0 hr. 12 min. 30-05 sec.

Equivalent interval of mean time, . 0 hr. 12 min. 28 sec.
L.M.T. corresponding to G.M.N., October,
2nd, 9 hrs. 14 min. 20-30 sec.

.-. L.M.T. required, . 9 hrs. 01 min. 52-3 sec.

In this case, since the instant precedes G.M.N., we must subtract the
computed interval of mean time from the L.M.T. corresponding to G.M.N.

Comparison of the Preceding Methods. — As it is a most
important thing that the student should thoroughly
grasp the principles involved in the transference of time
from one system of time measurement to the other, it
is a good exercise for him to master both the first method
given and the alternative method in each of the preceding
cases. The first method, however, involves less thinking
and is more mechanical than the other, so that it is the
method generally adopted and the one probably most
suited for ordinary computations.

Determination of the Local Mean Time of Transit of a Known
Star across the Meridian. — One very important application
of the preceding work is the calculation of the time of
transit of a known star across the meridian, or, as it is
commonly termed, the time of culmination.

The Nautical Almanac supplies us with a table of the
right ascensions and declinations of the principal stars
in the sky, and it has been shown in Chapter II. that
the R.A. of a star, expressed in time, is the sidereal time

TIME. 63

at the moment when the star is on the meridian. Thus
the problem is simply that of determining the' L.M.T.
corresponding to the sidereal time measured by the right
ascension of the star. This we may do by one of the
methods we have been considering.

EXAMPLE. — Find the time of culmination of a Tricing.
Aust. on the evening of August llth, 1913, at a place in
South Australia whose longitude is 139° 20' E., the time to
be measured in the standard time of the meridian 9 hrs.
30 min. E.

G.S.T. of G.M.N., August 17th, . . 9 hrs. 41 min. 02 sec.
.•. L.S.T. of L.M.N. at place in longitude

139° 20' E. computed as in previous

work, 9 hrs. 39 min. 30-45 sec.

R.A. of a Triang. Aust. = L.S.T. at time

of culmination, . . . .16 hrs. 39 min. 31 sec.
.•. interval of sidereal time elapsed since

L.M.N. , 7 hrs. 00 min. 00-55 sec.

Equivalent interval of mean time, . . 6 hrs. 58 min. 51-74 sec.

This, therefore, would be the L.M.T. at
time of culmination.
Difference between L.M.T. and time of the

standard meridian, 0 hr. 12 min. 40 sec.

•'. Standard time at culmination, . . 7 hrs. 11 min. 31-7 sec.

Time of Transit of the First Point of Aries. — In the
preceding work we have adopted the usual practice of
effecting the change from sidereal to mean or vice versa
by means of the column in the Nautical Almanac giving
the G.S.T. at G.M.N. But on page 3 of each month there
is given another column tabulating for each day in the
year the G.M.T. of transit of the First Point of Aries,
which may also be used for similar transformation of time .
As this instant indicates the beginning of the sidereal
day, the column might be appropriately headed, the
G.M.T. at sidereal noon.

64 ASTRONOMY FOR SURVEYORS.

Given the G.M.T. of Transit of the First Point of Aries, to
determine the L.M.T. of Transit at a Place in any other
Longitude.

The sidereal clock, as we have seen, is always gaining
on the clock keeping mean solar time, at the rate of
9-8565 seconds per mean solar hour, or at the rate of
9-8296 seconds for each sidereal hour. Now the G.M.T.
of transit of the First Point of Aries is the reading of the
mean time clock when the sidereal clock reads 0 hr. It
is the difference between the readings of the two clocks
at this instant. As the sidereal clock is gaining on the
other this difference will get less as the time increases.
Now, at a place in West longitude the transit of the
First Point of Aries will take place after an interval of
time measured in sidereal hours, minutes, and seconds
by dividing the longitude by 15. Thus, when this transit
occurs the mean time clock will not be so far ahead of
the sidereal clock as it was at Greenwich, and the Green-
wich reading of the mean time clock will have to be
diminished by subtracting 9-8296 seconds for each hour
of longitude.

This reasoning assumes that, whilst different clocks at
various places on the earth's surface will have different
readings according to the longitude, the difference between
the readings of the sidereal and mean time clocks at any
place is the same all over the world at the same instant.
This must be so according to the reasoning by which
we have established the rules for determining the local
mean and sidereal times at a place A, having given those
at a place B. For we should alter both the sidereal and
mean times at B by the same amount, depending on the
difference of longitude between B and A, in order to
find the corresponding times at A.

Accordingly we get the Nautical Almanac rule for finding
from the tables the time of transit of the First Point of
Aries at any place. " If the place of observation be not

TIME. 65

on the meridian of Greenwich, the mean time must be
corrected by the subtraction of 9-8296 sec. for each hour
(and proportional parts for the minutes and seconds) of
longitude, if the place be to the West of Greenwich ;
but by its addition, if to the East/'

EXAMPLE. — On August 1st, 1914, the G.M.T. of transit
of the First Point of Aries is 15 hrs. 20 min. 28-63 sec.
Compute the local time of transit on the same day (a) at a
place in longitude 57° 33' 28" West, (b) at a place in the
same longitude East.

(a) 57° 33' 28" is equivalent in time to 3 hrs. 50 min. 13-87 sec.

3 x 9-8296 - 29-488

50 x -1638 = 8-190

13-87 x -0027 - -037

37-715, say 37-72 seconds.

Therefore, for a place in West Longitude, we must subtract this from
the 15 hrs. 20 min. 28-63 sec., giving 15 hrs. 19 min. 50-91 sec. as the L.M.T.
of transit of the First Point of Aries.

(b) For a place in East Longitude we must add the 37-72 seconds, giving
15 hrs. 21 min. 06-35 sec. as the L.M.T. of transit in this case.

EXAMPLE. — Given that the G.M.T. of transit of the First Point of Aries
on August 30th is 13 hrs. 26 min. 27-26 min. Find the G.M.T. of transit
on August 31st. Find also the local mean time of transit at a place in
longitude 45° W.

Ans. 13 hrs. 22 min. 31-35 sec.
and 13 hrs. 25 min. 57-77 sec.

Given the L.S.T. at any Place and the G.M.T. of Transit of
the First Point of Aries on the same day, to determine the L.M.T.

The local sidereal time measures the interval in sidereal
hours since the transit of the First Point of Aries over
that meridian. By turning this, therefore, into mean
time hours we get the interval since the transit in mean
time hours. But we have just seen how we may calculate
the L.M.T. of transit of the First Point of Aries from the
information in the Nautical Almanac. The addition of
the two results will give us the L.M.T. required. The rule
of procedure, therefore, may be expressed : — Turn the

66 ASTRONOMY FOR SURVEYORS.

given sidereal time into mean time and add it on to the
computed L.M.T. of transit of the First Point of Aries.

As the transit of *¥» may take place at any time of the
day, some care is necessary in selecting the right transit,
as is illustrated in the following example :—

EXAMPLE. — Given that the L.S.T. at Mount Hamilton
is 22 hrs. 02 min. 47-02 sec. on October 2nd, 1913, the
longitude of the place being 121° 38' 43-35" West, find the
corresponding L.M.T.

Looking up in the Nautical Almanac the G.M.T. of transit of the First
Point of Aries on October 2nd we find it is 11 hrs. 15 min. 45-49 sec. This
is very near midnight, and the L.M.T. of transit will not be very different,
If we were to add 22 hours on to this it will clearly carry us over into the
next day, October 3rd, so that the transit we must select to work from,
is that on October 1st.

G.M.T. of transit of «Y» on October 1st, 11 hrs. 19 min. 41-39 sec.

Allowance for longitude, to be subtracted, 0 hr. 1 min. 19-71 sec.

L.M.T. of transit of <y> on October 1st, 11 hrs. 18 min. 21-68 sec.
Mean time equivalent to 22 hrs. 02 min.

47-02 sec. sidereal, 21 hrs. 59 min. 10-32 sec.

.•. L.M.T. required = October 2nd, . 9 hrs. 17 min. 32 sec.

Given the Sidereal Time at Mean Noon at Greenwich to compute
the Mean Time at the next Transit of the First Point of Aries.

The Nautical Almanac Columns, one giving the sidereal
time at mean noon and the other the mean time of transit
of the First Point of Aries, may readily be deduced one
from the other.

Thus, suppose the sidereal time at mean noon is denoted
by s. Then at noon s sidereal hours have elapsed since
<r> was on the meridian, and, therefore, in 24 — s sidereal
hours *Y> will again be on the meridian.

If we express 24: — s sidereal hours in mean solar
time, the result will clearly represent the number of
mean solar hours that have then elapsed since noon, and

TIME. 6t

will consequently represent the mean time at the next
transit of <Y^ .

For example, on November 1st, 1913, the sidereal
time at Greenwich mean noon is 14 hrs. 40 min. 40-14 sec.
Subtracting this from 24 hours, we get 9 hrs. 19 min.
19-86 sec. Turning this into mean solar time, the result
is 9 hrs. 17 min. 48-23 sec., which, therefore, represents
the mean time at the next transit of <Y» .

The converse problem may be dealt with in a similar
way.

EXAMPLE.— On October 28th the G.S.T. at G.M.N. is 14 hrs. 23 min.
56-95 sec. Find the mean time of the next transit of W .

Ans. 9 hrs. 34 min. 28-67 sec.

Nautical Almanac Data with regard to Time. — In the
Nautical Almanac on pages 1, 2, and 3 for each month,
various data are given that are useful in time computa-
tions. The sidereal time at Greenwich mean noon, the
mean time of transit of the First Point of Aries, and the
equation of time both for mean and apparent noon,
with its rate of variation, are given in each case for every
day in the year. In addition, the sun's right ascension
is given both for mean and apparent noon. These tabu-
lated results are not all independent, and it is good
practice for the student to take a Nautical Almanac
and deduce certain of the tabulated values from others
that are given. Here are a few of the exercises that may
be practised in this way.

1. From the sidereal time at mean noon on one day
compute its value for the next day.

2. From the sidereal time at mean noon find the mean
time of the next transit of the First Point of Aries.

3. From the mean time of transit of the First Point
of Aries determine the sidereal time at mean noon on the
same day.

4. From the R.A. of the sun at mean noon, and the

68 ASTRONOMY FOR SURVEYORS.

equation of time, with their rates of variation, deduce
the sidereal time at mean noon, and the R.A. of the sun
at apparent noon.

5. From the sidereal time and the sun's R.A. at mean
noon, deduce the equation of time.

EXAMPLES.

1. Express in sidereal time the following intervals of mean solar time :• —
(1) 16 hrs. 15 min. 23 sec., (2) 9 hrs. 17 min. 18-4 sec., and (3) 17 hrs. 52 min.
33-5 sec.

Ans. (1) 16 hrs. 18 min. 3-2 sec.

(2) 9 hrs. 18 min. 49-95 sec.

(3) 17 hrs. 55 min. 29-69 sec.

2. Express in mean solar tune the following intervals of sidereal time : —
(1) 13 hrs. 22 min. 17 sec., (2) 21 hrs. 35 min. 15-5 sec., and (3) 8 hrs. 55 min.
39-7 sec.

Ans. (1) 13 hrs. 20 min. 05-56 sec.

(2) 21 hrs. 31 min. 43-3 sec.

(3) 8 hrs. 54 min. 11 -94 sec.

3. In longitude 148° 15' E., what is the local mean time corresponding
to September 22nd, 4 hrs. 30 min. p.m., standard time of the 150th meridian
East of Greenwich ? Find also the corresponding Greenwich mean time.

An*. (1) 4 hrs. 23 min. p.m.
(2) 6 hrs. 30 min. a.m.

4. Convert Perth apparent time, December 3rd, 4 hrs. 15 min. 20-3 sec.
to sidereal time ; also Perth sidereal time, December 3rd, 20 hrs. 26 min.
16-7 sec., to Western Australian standard time (Time of 120th meridian).

Given longitude of Perth, . . 7 hrs. 43 min. 21-7 sec. E.
Sidereal time as G.M.N., Dec. 3rd, . 16 hrs. 47 min. 32-0 sec.
„ Dec. 2nd, . 16 hrs. 43 min. 35-5 sec.

Equation of time G.M.N., Dec. 3rd, 10 min. 10-1 sec. to be added to

mean time.
„ „ Dec. 2nd, 10 min. 33-5 sec.

Ans. Sidereal time —

20 hrs. 52 min. 03 sec.
L. Standand time —
3 hrs. 56 min. 02-1 sec.

TIME. 69

5. Given that the sidereal time at Greenwich mean noon is 14 hrs. 40 min.
40-14 sec., find the mean time of the next transit of the First Point of
Aries.

Ana. 9 hrs. 17 min. 48-23 sec.

6. Given that the mean time of transit of the First Point of Aries at
Greenwich is 11 hrs. 19 min. 41-39 sec., compute the sidereal time at Green-
wich mean noon on the same day.

Ans. 12 hrs. 38 min. 26-95 sec.

7. The right ascension of a star being 20 hrs. 24 min. 13-72 sec., compute
the local mean time of its culmination at Madras (longitude 80° 14' 19-5" E.)
on September 6th, the sidereal time at Greenwich mean noon on that date
being 11 hrs. 2 min. 21-45 sec.

Ans. 9 hrs. 21 min. 12-8 sec.

8. Convert 22 hrs. 22 min. 44-58 sec. sidereal time at Greenwich, January
20th, 1913, into mean time, given that the mean time of transit of the
First Point of Aries on January 19th is 4 hrs. 6 min. 14-36 sec.

Ans. 2 hrs. 25 min. 18-96 sec.

9. Find the mean local time corresponding to 5 hrs. 17 min. 32 sec. sidereal
time at Moscow (longitude 37° 34' 15" E.), given that the sidereal time
of Greenwich mean noon on the same day was 23 hrs. 54 min. 52 sec.

Ans. 5 hrs. 22 min. 11 sec

10. Find the standard time of culmination of a Centauri at Adelaide
on June 1st, 1914, R.A. = 14 hrs. 33 min. 49 sec., longitude = 9 hrs. 14 min.
20-3 sec. Standard meridian 9 hrs. 30 min. E. G.S.T. at G.M.N. on the
same date — 4 hrs. 36 min. 30-1 sec.

Ans. 10 hrs. 12 min. 51-3 sec.

1 1 . Find the local mean time of the transit of £ Crucis over the meridian,
at a place in longitude 11 hrs. 30 min. E. on the 10th May, 1913. Transit
First Point of Aries, G.M.T., 9th May, 20 hrs. 49 min. 48-44 sec. ; star's R.A.,
12 hrs. 10 min. 33-08 sec.

Ans. 9 hrs. 00 min. 14-9 sec.

12. The mean time of transit of the First Point of Aries for January
21st, 1911, is given in the Nautical Almanac as 4 hrs. 00 min. 24-79 sec.
For the same date the R.A. of a Leonis is given as 10 hrs. 03 min. 38-76 sec.
Find the exact local mean time when a Leonis passed the meridian of a
place in longitude 135° E.

Ans. 2 hrs. 03 min. 53-13 sec. a.m.,
Januarv 22nd.

ASTRONOMY FOR SURVEYORS.

13. Compute the local sidereal time at noon by standard time at Adelaide

on October 24th, 1914, given
Longitude of Adelaide, .
Longitude of standard meridian,
G.S.T. at G.M.N., October 23rd,

9 hrs. 14 min. 20-30 sec. E.
9 hrs. 30 min. E.
. 14 hrs. 04 min. 14-18 sec.

Ans. 13 hrs. 50 min. 57-40 sec.

14. In the forenoon of August 1st, 1914, at Melbourne, longitude 9 hrs.
39 min. 54 sec. E., a mean time chronometer was compared with a sidereal
clock known to be 14-6 seconds fast on true local sidereal time. It was
found —

Time by sidereal clock, .
Time by chronometer,

The data in the appended table is taken from the Nautical Almanac : —

8 hrs. 18 min. 09-00 sec.
11 hrs. 41 min. 34-32 sec.

GREENWICH MEAN NOON.

Date— 1914.

Apparent R.A. of
Sun.

^^-/"k^SSe^
One Hour. , frQm Meftn T-me

Variation in
One Hour.

July 31, .
Aug. 1, .
Aug. 2, .

His. Mins. Sees.
8 39 18-14
8 43 11-80
8 47 04-84

Sees. i Wins. Sees.
9-749 6 14-54
9-723 6 11-65
9-697 6 08-13

Sees.
0-108
0-134
0-159

Determine

(a) The sidereal time at Greenwich mean noon, August 1st.

(b) The R.A. of the sun at apparent noon, August 1st.

(meridian 10 hrs. E.).

Ans. (a) 8 hrs. 37 min. 0-16 sec.

(b) 8 hrs. 43 min.. 12-80 sec.

CHAPTER VI.

THE LOCATION OF OBJECTS ON THE CELESTIAL
SPHERE.

IN order that the surveyor may pick out and observe
a particular star with a theodolite, it is frequently neces-
sary, more especially when he wishes to make the obser-
vation in daylight or evening twilight, that he should
know the altitude and azimuth of the star at the given
time. From the Nautical Almanac he obtains its right
ascension and declination, and from these data he has
to compute altitude and azimuth. In this chapter we
will deal with this problem and show how, given the
position of a star in one system of co-ordinates we may
determine its co-ordinates in another.

A . Knowing the Latitude and Time at the Place of Observation
and the Right Ascension and Declination of a particular Star, it is
required to determine its Altitude and Azimuth.

In Fig. 16, let P be the pole, S the star, Z the zenith,
A Z P B the plane of the meridian.

Draw the great circle through Z and S to intersect the
horizon in H.

If we know the local mean time we can compute the
corresponding sidereal time by the methods of the last
chapter. But we have seen that the right ascension of
the star is the same thing as the sidereal time at the
moment of the star's transit across the meridian. Con-
sequently the difference between the sidereal time at the
instant of observation and the right ascension of the
star gives the interval in sidereal time between the
moments of the star's transit across the meridian and of

72 ASTRONOMY FOR SURVEYORS.

observation — that is to say, it gives, when turned into
degrees, minutes and seconds, the hour angle of the
star S P Z. If the sidereal time at the moment of observa-
tion is less than the right ascension of the star, the differ-
ence measures the angle S P Z towards the East of the
meridian, if the right ascension is the less, the angle is
measured toward the West.

Thus, in the spherical triangle Z S P, we know Z P,
the complement of the latitude, and S P, the polar dis-
tance of the star which is the complement of the declin-
ation, and the included angle Z P S.

From these data we can compute the third side Z S,

Fig. 16.

which is the zenith distance of the star, or the com-
plement of the altitude, and the angle S Z P, which
determines the azimuth.

Calling the angles of the spherical triangle Z, P, and
S respectively, the formulae applicable to the solution
of a spherical triangle, having given two sides and the
included angle, are

LOCATION OF OBJECTS ON CELESTIAL SPHERE. 73

From these equations we compute the angles S and Z.
Then, to determine S Z, we have

sin P sin S P

sin SZ

sin Z

EXAMPLE. — At a place in South Australia in longitude
9 hrs. 14 min. E., latitude 32° 35' S., it is required to deter-
mine the altitude and azimuth of Achernar at 7 p.m. standard
time on December 1st, 1913. The R.A. of Achernar is
1 hr. 34 min. 33 sec., and its declination South is 57° 40' 33".

The standard time of South Australia is that of the meridian 9 hrs. 30 min.
E.

The Greenwich time corresponding to 7 p.m. standard time on December
1st is thus 21 hrs. 30 min. on November 30th.

Therefore, the interval of time which has elapsed since Greenwich noon
on November 30th is 21 hrs. 30 min. of mean time, equivalent to 21 hrs.
33 min. 31-9 sec. of sidereal time.

From the Nautical Almanac, the sidereal time at Greenwich noon on
November 30th is . . . . . .16 hrs. 35 min. 0*3 sec.

Difference due to longitude, ... 9 hrs. 14 min. 0 sec.

Local sidereal time at Greenwich noon. . 1 hr. 49 min. 0-3 sec.

Interval of sidereal time since elapsed. . . 21 hrs. 33 min. 31-9 sec.

Local sidereal time required, . . .23 hrs. 22 min. 32-2 seo.

This gives us the sidereal time at the instant of observation.
But the R.A. of Achernar is 1 hr. 34 min. 33 sec.

Thus Achernar lies 21 hrs. 47 min. 59-2 sec. to the West of the meridian,
or 2 hrs. 12 min. 0-8 sec. to the East.

Multiplying this by 15, we get the hour angle of the star as 33° 0' 12"
to the East.

Referring now to Fig. 16, we have

Z P = co-latitude = 57° 25'

P S = complement of declination = 32° 19' 27"

P - 33° 0' 12"

cos | (Z P - S P) = cos 12° 32' 46-5", . . 9-9895036
cot £ P = cot 16° 30' 6", . . 10-5283488

10-5178524

cos | (Z P + S P) = cos 44° 52' 13-5", . . 9-8504650

tan |(S 4- Z), 10-6673874

74 ASTRONOMY FOR SURVEYORS.

.-. J (S + Z) = 77° 51' 40".

sin i (Z P - S P) = sin 12° 32' 46-5", . . 9-3369150
cot £ P = cot 16° 30' 6", . . 10-5283488

9-8652638
sin £ (Z P + S P) = sin 44° 52' 13-5", . . 9-8485005

tan | (S - Z), . . . . . 10-0167633

.-. £ (S - Z) = 46° 6' 20"

.-. Z = 31° 45' 20".

Thus the star lies in the direction 31° 45' 20" East of South.

To find its altitude,

sin P = sin 33° 0' 12", .... 9-7361477
sin S P = sin 32° 19' 27", . . 9-7281173

9-4642650
sin X = sin 3r 45' 20". 9-7212303

sin S Z, -. 9-7430347

... SZ = 33° 36' 1".

Therefore, the altitude of the star is the complement of this, or 56°
23' 59".

Very commonly for such calculations it is sufficient to compute the
position of the star to the nearest minute, and in that case five-figure log-
arithms are sufficient.

/> . Having observed the Altitude and Azimuth of a Star, the
Time of Observation being noted, it is required to determine its
Right Ascension and Declination.

The latitude and longitude of the place of observation
are supposed known.

Then in the figure, Z being the zenith point, P the pole,
and S the star, as before.

In the spherical triangle Z S P, Z P is known, being
the co-latitude ; Z S, the zenith distance, is also known,
and the angle S Z P, which the vertical plane passing
through the star makes with the meridian.

Thus we know two sides and the included angle, and
the triangle may be solved to find S P and the angle
SPZ.

LOCATION OF OBJECTS ON CELESTIAL SPHEEE. 75

The formulae to be used are those of the preceding
problem.

*1*#&£$&3R»

sinSP-

sin | (ZP+ZS)
sin Z . sin Z S
sinP '

The angle S P Z, being turned into hours, minutes, and
seconds, at the rate of 15° for one hour, measures the
sidereal time that will elapse before S comes to the meri-

Fig. 17.

dian if S is to the East, or the interval of sidereal time
since S was on the meridian if it is to the West.

But the right ascension of the star is the sidereal time
when it is on the meridian.

Therefore, to obtain the right ascension of the star,
add the time value of the angle S P Z to the local sidereal
time at the moment of observation if the star is to the
East of the meridian, and subtract it if the star is to the
West.

The declination of the star is, of course, the complement
of the computed polar distance S P.

76 ASTRONOMY FOR SURVEYORS.

C. Having computed the Altitude and Azimuth of a Star for a
Given Time of Observation, it is required to determine its Approxi-
mate Position at some Short Interval of Time afterwards.

When a surveyor is preparing for daylight observations
of a star, it will be generally necessary for him to take at
least two readings of its position. To give him time to-
read the verniers and reverse the instrument before taking
the second observation, he requires to know the altitude
and azimuth of the star at an interval of five or ten
minutes after the first reading.

The computation for the second position may, of course,
be made in precisely the same way as we have already
done for the first, in which case several of the logarithms
already taken out will be useful for the calculation.

But it is rather shorter to make use of the following
formulae : —

If x denotes the slight increase in the hour angle S P Z
(to be reckoned negative if the angle is decreasing), y
and z the corresponding small increases in the zenith
distance Z S, and the azimuth angle P Z S respectively.
Then

y= sin PS sin PSZ .x. . . (1)

cot^SZ^,
sinZS

The values of P S Z and Z S to be used in the equations
being those found in the first calculation.

To establish the formulae, let ABC (Fig. 18) be a
spherical triangle. Then if b and c remain unchanged,
we require to find the small changes y and z in a and B
respectively if the angle A is increased by a small amount
x.

By the ordinary formulae for spherical triangles we
have

cos a = cos b cos c -f- sin b sin c cos A
and cos (a + y) = cos b cos c + sin b sin c cos (A -f- x)

LOCATION OF OBJECTS ON CELESTIAL SPHERE. 77
Subtracting gives

cos a cos y — sin a sin y — cos a = sin b sin c
(cos A cos a:— sin A sin x— cos A).

Now, if x and y are very small, we can, if they are
measured in circular measure, replace sin x and sin y
by x and y respectively, and put cos x, cos y each equal
to unity. Doing this, we get

- y sin a = — sin b sin c sin ^4 . #.
Putting sin c sin A = sin C sin a, this becomes
y = sin b sin C . #, '

which is the first formula given.

Since we have here simply the ratio
of y to .r, the result will hold good in
whatever system of measurement y ,
and x are expressed, provided they cj
are both measured in the same system,
both in degrees or both in circular
measure.

Further, by the law of sines,

sin(B+z) sin (A -fa)

sin b

sin (a+y)'

Expanding and substituting as before,
we get

(sin B + z cos B) (sin a + y cos a) = sin b (sin A -f x cos A)
and sin B . sin a = sin b . sin A.

.-. substracting, and neglecting the product of two
small quantities y and z,

z sin a cos B -f y cos a sin B = x sin b cos A.

78 ASTRONOMY FOR SURVEYORS.

Putting x= . , . -

sm b sin C

/cos A cos B cos C

z sin a cos B = y ( - — sin B cos a ) = — y : — ~ — -

VsmC sm C

... z = — y . — — , which is the second formula,
sin a

To illustrate the application of the formulae we will extend the scope
of the example already worked out in Section A of this Chapter, and
compute the position of Achernar 5 sidereal minutes after 7 p.m.

From the previous work the angle P S Z = 123° 58', P S = 32° 19',
Z S = 33° 36'.

sin PS, . . . . • . . . 9-72803
sinPSZ, . 9-91874

-44338, . 1-64677

In this example the hour angle of the star is measured to the East, and,

therefore, x is negative, and = — 5 minutes of time = — 1° 15' of arc.
.-. y = - -44338 x 75' = - 33'.
.-. The new altitude is 56° 24' + 33' = 56° 57'.

cot P S Z. 9-82844

sin Z S, 9-74303

1-2173, . ... . 0-08541

and cot P S Z is negative, .-. z = 1-2173 x (— 33') = — 40'.
.-. The new azimuth is 31° 45' — 40' = 31° 5' East of South.

If results are only required to the nearest minute, the
above method is quite sufficient, provided the small
differences are not much more than 2 degrees of arc.

EXAMPLES.

1. Compute to the nearest minute of arc the altitude and azimuth of
Sinus (dec. = 16° 35' South, R.A. = 6 hrs. 41 min.) at a place in latitude
31° 57' South at 12 hrs. sidereal time.

Ans. Azimuth = 260° 51'.
Altitude 17° 12'.

2. Compute the* altitude and azimuth of Sirius 10 sidereal minutes later
than in I .

Ans. Azimuth = 259° 38'
Altitude = 15° 7'.

LOCATION OF OBJECTS ON CELESTIAL SPHERE. 79

3. At a place in latitude 28° South at 1 hr. 37 min. sidereal time, th»
altitude of Canopus is observed as 33° 3' and its azimuth as 136° 44'. Com-
pute the R.A. and dec. of the star.

Ans. R.A. = 6 hrs. 21 min.

58 sec.
Dec. = 52° 38' 48" S.

4. What is the angular distance between the stars A (R.A., 4 hrs. 23 min.
53 sec., Dec., 16° 04' 25" N.) and B (R.A., 2 hrs. 54 min. 34 sec., dec., 40°

08'03"N.)?

Ans. 30° 54' 14".

5. Find the angular distance between A (R.A., 19 hrs. 42 min. 11 sec.,
dec., 8° 23' 52" N.) and B (R.A., 22 hrs. 47 min. 41 sec., dec., 30° 33'
17" N.).

Ana. 59° 06' 04".

6. If the N. dec. of a star is 40°, show that the number of hours in the
sidereal day during which it will be below the horizon of a place which has
latitude 30° N. is 8-136.

CHAPTER VII.

ASTRONOMICAL AND INSTRUMENTAL CORREC-
TIONS TO OBSERVATIONS OF ALTITUDE
AND AZIMUTH.

Parallax. — The fixed stars are so distant from us that
their directions always appear to be the same, no matter
from what point upon the earth's surface they are observed.
Even with our most refined instruments no difference can
be detected, because their distance is practically infinitely
great in comparison with the diameter of the earth. But
with the members of our own system, the sun, the moon,
and the planets, we are dealing with bodies incomparably
nearer to us, and their relative positions amongst the
fixed stars of the sky are not precisely the same when
viewed from different places. It is, therefore, essential
that their registered right ascensions and declinations
should be referred to some definite point upon the earth,
in order that they may be available to all observers.
The point selected is the earth's centre, because, having
observed the direction of a planet from any station on
the earth's surface, it is an easy matter to deduce its
position as it would appear at the earth's centre, and
conversely if the position of the star is tabulated as it
would be seen from the centre of the earth we may
readily find its position as seen from any place on the
earth's surface. The selection of the earth's centre as the
imaginary place of observation greatly simplifies the com-
putations, and consequently most astronomical obser-
vations of bodies in our own solar system are reduced

ASTRONOMICAL AND INSTRUMENTAL CORRECTIONS. 81

so as to show what the result would be if the observation
could have been at the centre of the earth. The registered
right ascensions and declinations of the Nautical Almanac
are those the different bodies would have if viewed from
the earth's centre.

The difference between the directions of a heavenly
body as seen from the earth's centre and as seen from the
place of observation is known as its Parallax. ^

Thus, as in Fig. 19, if S is the sun or planet observed,

d ?

Fig. 19.

A the point of observation, and 0 the earth's centre, the
parallax of the body is the angle A S 0, the difference
in the directions of A S and OS. If A Hx is the direction
of the horizontal at A, the altitude of S is the angle S A H1.
If 0 H2 is drawn parallel to A Hj, then the difference of
the angles S O H2 and S A Hj = the difference of the

82 ASTRONOMY EOR SURVEYORS.

angles SOB and SAB which = the angle A S O.
Thus, if we call p the parallax, p= angle A S 0= S 0 H2
- S A Hj. Clearly the angle S 0 H2 is always greater
than the angle S A Hj.

If z denotes the zenith distance of S as observed from
A, r the earth's radius 0 A, and d the distance 0 S, then,

mn /7) Y

From the triangle A O S, - — = - .

sin z d

If the body is observed on the horizon — that is to say,
if z= 90° — the corresponding value of p is called the
horizontal parallax. Call this P.

Then sin P = *,.

Therefore, sin p = sin P . sin z.

Since p and P are very small, except in the case of the
moon, whose parallax sometimes exceeds 1°, we may
substitute the angles for their sines and write

p = P sin z.

The horizontal parallax of the moon and principal
planets is given in the Nautical Almanac for every day
in the year, and that of the sun at intervals of 10 days.
The parallax for any other altitude is given by the above
simple formula.

Parallax is greatest when the body is in the horizon,
and diminishes with the altitude until it becomes nothing
when the body is in the zenith.

We see from Fig. 19 that the effect of the parallax
upon a celestial object is to make its altitude appear
less when observed from A than it would be if seen from
O. Consequently, when reducing observations to the
earth's centre, we must add the correction for parallax
observed to the altitude, or

True altitude = observed altitude -f parallax.

ASTRONOMICAL AND INSTRUMENTAL CORRECTIONS. 83

Parallax has no effect upon the azimuth of an object
in the sky ; the correction is made to altitude only.

This statement is strictly correct only when the earth
is regarded as a perfect sphere. If 'the spheroidal form
of the earth is taken into account there will be parallax
in azimuth as well as in altitude. Even then, however,
the correction in azimuth is too small to be worth con-
sidering except in the case of certain special lunar obser-
vations.

The horizontal parallax of the sun ranges between
8-65 and 8-95 seconds. At an altitude of 60° its parallax
is reduced to half of this.

Atmospheric Refraction.

When a ray of light passes
from one medium into a
denser medium as from air
into water or from air into

glass, it is bent or refracted
towards the normal to the

bounding surface. Thus, as //////////////?/;(//////////////

in Fig. 20, if a ray of light
passes from the medium A to
a denser medium B, travers-
ing the path P Q R, the re-
fracted ray Q R will always
make a smaller angle with
the normal to the separating
surface than the incident
ray P Q. The direction of bending is always such
that the bent or refracted ray lies in the same
plane as that passing through the incident ray P Q
and the normal Q N. The law governing the amount
of bending is that the ratio between the sines
of the angles P Q N and R Q M is constant for these

Fig. 20.

84 ASTRONOMY FOR SURVEYORS.

particular media and the value of this ratio is known as
the coefficient of refraction.

Similarly, when a ray of light from a celestial body
reaches the atmosphere surrounding the earth, it is
bent slightly out of its original path. If the atmosphere
were a uniform homogeneous medium with a definite
upper surface it would be comparatively easy to deter-
mine the precise amount of bending of the ray. But the
density of the atmospheric air diminishes with the height
above the earth's surface. Consequently a ray from a
star S (Fig. 21), when it reaches the upper limit of the

Fig. 21.

earth's atmosphere at A, is only very slightly bent, but
the amount of bending gradually increases as it passes
into the lower and denser layers of air. Its path from A
to an observer on the earth's surface at 0 is thus a curve,
and the ray ultimately reaches the observer, so that it
appears to him to come in the direction of 0 S1. Thus,
the observer sees the star apparently at S1 in the celestial
sphere, whereas in reality the star is at S. The effect is
that the star is apparently raised above its true position,
and its apparent altitude is greater than the true altitude

ASTRONOMICAL AND INSTRUMENTAL CORRECTIONS, 85

if it could be observed from 0 with no intervening atmo-
sphere. The observed altitude of a celestial body must,
therefore, be corrected in order to deduce its true altitude,
the correction being always subtracted from the observed
altitude. The amount of bending of the ray varies
somewhat with the pressure and temperature of the air,
but it is greatest for stars on the horizon, and gradually
decreases to nothing for a star in the zenith. For a body
on the horizon the mean value of the correction is 33'—
that is to say, a star will be just visible on the horizon
when it is really 33' below it. Thus the sun, whose dia-
meter is about 32', is visible just above the horizon when
it is in reality just below it.

It will be seen from the figure, since the refracted ray
always lies in the plane containing the incident ray S A,
and the normal to the spherical bounding surface at A,
that S and S' will lie in the same plane as the vertical
at 0. This means that refraction produces its effect
entirely in altitude, and has no influence upon the apparent
azimuth of a heavenly body. Thus no correction in azi-
muth is necessary on account of refraction.

As we do not know the exact laws which govern the
pressure and temperature of the earth's atmosphere at
different heights, nor even the distance to which it extends
around the earth, no satisfactory computation of the
amount of refraction at different altitudes can be made
from theoretical considerations alone. By making different
assumptions as to the character of the earth's atmosphere
various formulae have been derived, but as their demon-
stration generally requires mathematics of a rather
advanced character, we shall not attempt the problem
here. In any case, as we cannot be sure of the correctness
of the assumptions that have to be made in order to
derive the formula, the values of the constants used have
to be obtained and checked from actual observations.
There are various ways by which the amount of refraction

86 ASTRONOMY FOR SURVEYORS.

at different altitudes may be actually measured, and for
practical purposes that formula is selected which best
fits the results of such measurements.

The formula that has found most favour, and which
has been most used by astronomers for this purpose, is
that of Bessel,

r= A (B Z)M TN cot a,

where a = the apparent altitude,

r= the amount of refraction in seconds of arc,
B, a factor depending on the height of the

barometer,
t, a factor depending on the reading of the

thermometer attached to the barometer,
T, a factor depending on the reading of a ther-
mometer so exposed as to give the
temperature of the external air.

A, M, and N are factors depending on the altitude of
the celestial body.

When suitable values are given to the different factors,
this formula can be made to fit in with the results of
actual observations on refraction with great precision,
and where great accuracy is required this is the formula
that is most generally adopted. To use the formula it
is, of course, put into the logarithmic form —

log r = log A+ M (log B -f log t) + N log T + log cot a,

and the values of M, N, log A, log B, log t, and log T are
obtained from appropriate tables. Such a table is published
in Chambers' Mathematical Tables.

The constants M and N in the above formula do not
differ sensibly from unity if the altitude is considerable.
If these are taken each= 1, the formula may be put
into a form which makes the application of tables much
simpler. For the values of B, t, and T are each unity for
certain particular values of the barometric height, and
for certain special temperatures of the attached and

ASTRONOMICAL AND INSTRUMENTAL CORRECTIONS. 87

unattached thermometers. Consequently for this par-
ticular condition of the atmosphere, which we may take
as the standard condition, we have r = A cot a.

If now we denote by r1 the amount of the refraction
for any other temperature and pressure, we have —

^ = A . B t . T cot a,
... r1='BxtxTxr,

or refraction = the refraction for altitude a under the
standard or mean conditions multiplied by the factors
B, t, and T, depending on the height of the barometer
and the temperatures recorded by the attached and
unattached thermometers.

A table of refractions constructed for standard con-
ditions of the atmosphere is commonly termed a table of
mean refraction. With the aid of such a table and sub-
sidiary tables for B, t, and T, we may first of all find the
value of the " mean refraction " for the measured altitude,
then pick out the values of B, t, and T for the particular
conditions of the atmosphere, and the true refraction
— the mean refraction x B x t x T.

This is the method of determining the refraction most
commonly adopted for ordinary purposes, and gives accu-
rate enough results unless the altitude is very small. The
necessary tables are in Chambers' Mathematical Tables.

For many purposes, and more especially for high
altitudes, it is quite sufficiently accurate to use the value
of the refraction as given in the mean refraction table.
The refraction is always less than 1' if the altitude is greater
than 45°, and for zenith distances up to 20° the refraction
is practically 1" per 1°.

Corrections to Observations on Account of Residual
Instrumental Errors.

It forms no part of the purpose of this book
to enter upon a discussion of the construction of the
ordinary instruments of the surveyor and the methods

88 ASTRONOMY FOR SURVEYORS.

of adjustment. These are matters dealt with in
text-books on Surveying. It will be assumed that the
reader is acquainted with the construction of the sur-
veyor's transit theodolite and with the usual methods
of securing its accurate adjustment. But even when
the adjustments have been made with great care, there
commonly remain certain residual errors which affect
the accuracy of the celestial observations, and must be
taken into account if the best results are to be obtained.
Of these, the two most important are, (1) an error due
to the fact that the line of collimation of the telescope is
not accurately at right angles to the transverse axis
about which the telescope turns, and (2) an error produced
if this transverse axis is not absolutely horizontal. We
will consider the effect of each of these in turn.

The Effect of an Error of Gollimination. — Let us suppose
that the line of collimation of the telescope, instead of
being accurately at right angles to the axis about which
the telescope turns, is in error by a small angle c ; that
is to say, the telescope makes an angle 90° — c on one
side and 90° -fc on the other side with the axis. On
turning the telescope about the transverse axis, which is
adjusted so as to be horizontal, the line of collimation
would, if in accurate adjustment, trace out a vertical
plane passing through the zenith. But if in error, and the
line of collimation is not at right angles to the axis, then,
as it is plunged up and down, it will trace out a conical
surface and on the celestial sphere it will trace out a
circle parallel to a vertical circle through the zenith.
Thus, as in Fig. 22, if there were no collimation error the
line of collimation of the telescope would trace out the
great circle Z S' N, but if in error it will sweep out the
parallel small circle L S M. Now, suppose that the star
S is observed in such a telescope, and let S S' be an arc
of a great circle drawn at right angles to Z N. S S' = N M
= c the collimation error.

ASTRONOMICAL AND INSTRUMENTAL CORRECTIONS. 89

If we draw the great circle arc Z S, then Z S is the
true. zenith distance of the star. But the observed zenith
distance is Z S'. Similarly the correct azimuth is measured
by the angle H Z S, whereas the azimuth as read on the
instrument is H Z S'.

In the right-angled triangle S S' Z, S S' being denoted
by c, we have

cos S Z = cos S' Z cos c.

If c is very small, as should be the case if the instrument
is in decent adjustment, we may take cos c= 1, and,
therefore, practically S' Z = S Z, or no correction will

Is \

N M

Fig. 22.

usually be necessary to the observed zenith distance or
altitude.

Also, denoting by Z the angle S Z S', the error in
azimuth, we have

sin c = sin S Z . sin Z,

and since c and Z are both small, we may write
Z = c . cosec S Z,

or the error in azimuth = the collimation error multiplied
by the cosecant of the zenith distance.

The error in azimuth thus becomes very great if the
star is near the zenith, but is= c for a star on the horizon.

90 ASTRONOMY FOR SURVEYORS.

The following table shows the way in which the error
varies with the altitude of the star : —

Error in Azimuth corresponding to a Collimation
Error c for Various Altitudes of Object.

Altitude of Star, 0° 30° 60° 70° 80° 85° 89°
Error in azimuth, c M5c 2c 2-92c 5-76c ll-47c 57-3c

The Elimination of Instrumental Errors by Changing Face. —
Although we have in the preceding paragraph investi-
gated the effect of a given collimation error, it is very
seldom that the surveyor will need to take this error
into account, because in all important work the observa-
tions are taken in such a way as to eliminate its effects.
This is done by observing each angle twice, with the
vertical circle or face alternately to the left and to the
right. After the angle has been read once the telescope
is reversed in direction by turning about its horizontal
axis, and the whole of the upper part of the theodolite
is turned through 180° until the first object is again
sighted, and the angle is again read with the instrument
in this reversed position. The operation is commonly
referred to as " changing face/' and should be adopted
in all theodolite observations, as it gives a means both
effectual and simple of eliminating the chief instrumental
errors. An error in collimation will not affect the hori-
zontal angle between two objects if both are at the same
altitude, but if the altitudes are different, then if the
collimation error makes the measured angle a little too
great when the vertical circle is facing the left it will make
it just as much too small when the vertical circle faces
the right, and thus the "mean of the two readings gives
the correct result.

Now, when measuring the azimuth of a star, we have to
sight the telescope to a moving object, and it is not possible,
therefore, to exactly repeat the measurement because in

ASTRONOMICAL AND INSTRUMENTAL CORRECTIONS. 91

the interval of time taken in changing face the position
of the star is slightly changed. But it is characteristic
of all the more accurate methods of astronomical measure-
ment suitable for the surveyor, that reliance is never
placed upon one observation, but the methods are so
arranged that a series of observations can be made at
short intervals, the face of the instrument being alter-
nately changed from right to left, so that a mean may be
obtained from which instrumental errors are largely
eliminated.

The Error made if the Transverse Axis of the Telescope is
not truly Horizontal. — This error, just as that due to
collimation with which we have just dealt, may also

Fig. 23.

be largely eliminated by the method of changing face.
But in this case the elimination is not so perfect, and as
it is an easy matter by means of a striding level to actually
measure the departure of the axis from the horizontal
at each observation, it is frequently desirable to observe
the error and allow for it in the computation.

If the axis of the telescope is not truly horizontal, the
line of collimation, when the telescope is turned about
the axis, will not trace out a great circle in the sky passing
through the zenith, as it should do, but will trace out a
great circle inclined to the vertical. Thus in Fig. 23,
if N Z N1 denotes the great circle that would be traced

<J2 ASTRONOMY FOR SURVEYORS.

out in the celestial sphere if the axis were horizontal,
N S N1 denotes the circle actually traced out if the axis
is inclined at a small angle a. Let S be a star observed
with this telescope, and draw the great circle Z S M passing
through the zenith and the star.

The angle Z N S = a.

The actual observed altitude of the star is measured
by the arc N S, whereas the true altitude is given by
the arc M S.

Again, the azimuth of the star is actually measured
on the circle of the horizon from the point N, whereas
it should be measured from the point M. So that the error
in azimuth is the angular measure of the arc M N.

In the right-angled triangle N S M, the angle S N M
= 90°— a. Therefore, by Napier's rules, we have

sin N M = tan a . tan M S,
or, since both N M and a are small,

NM= a. tan MS.

That is to say, the error in azimuth = the error in level
multiplied by the tangent of the altitude of the star.
Again, by Napier's rules,

sin M S = sin N S . cos a,

and since a is small and cos a may be taken = 1 , it follows
that we may take M S = N S, which means that no
appreciable correction has to be made to altitude. The
error produced is practically in azimuth only.

The error in azimuth increases with the altitude of
the star. It is zero on the horizon, becomes = a for an
altitude of 45°, and is very great for stars near the zenith.

Error in Azimuth corresponding to a Level Error a in the
Axis for Various Altitudes of Object.

Altitude of star, 0° 30° 45° 60° 70° 80° 85° 89°
Error in azimuth, 0 0-58a a l-73a 2-75a 5-67a ll-43a 57-3a

ASTRONOMICAL AND INSTRUMENTAL CORRECTIONS. 93

Determination of the Level Error of the Axis by means of
the Striding Level. — In order to make practical applica-
tion of the correction just investigated, it is necessary
to actually measure the level error of the transverse axis
of the theodolite for each observation. This is readily
done by means of the striding level, a very sensitive
spirit level supported by two legs with V bearings at
the bottom, which can rest upon each end of the trans-
verse axis of the theodolite. The tube of the level is
marked off in divisions, the values of which are known
or may be readily determined by test. The graduations
read outwards from the centre towards both ends. To
eliminate errors of construction the readings should be
taken in pairs, the striding level being read first in one
position and then reversed on its bearings with each
observation. Both ends of the bubble are read on each

Fig. 24.

occasion, the observer standing so as to face the direction
in which the instrument is pointed. He reads first the
left-hand end, then the right, then reverses the level and
reads again.

Suppose, as in Fig. 24, the bubble extends from A to B,
0 being the centre of the graduations and C the middle
point of the bubble.

Then C B = half the length of the bubble

O_A + 0 B

OB-0 A

•94 ASTRONOMY FOR SURVEYORS.

This, therefore, measures the deflection of the centre
of the bubble from its normal position, and, when
multiplied by the value of 1 division of the level,
gives the angular measure of the deflection from the
horizontal.

Suppose that the readings of the left-hand and right-
hand ends of the bubble are l± and r± respectively before
reversal and 12 and r2 after reversal of the level on its
bearings. Then, according to the first reading, the error

I- — i\
of the axis is , and according to the second reading

^2 — ^2

— — — . Thus the mean determination is

We thus get the following rule, for finding the error in
level of the horizontal axis, after a series of striding
level readings taken in this way. Add up the left-hand
readings. Add up the right-hand readings. Subtract
the two sums and divide by the total number of readings.
The result is to be multiplied by the value of the level
graduation in seconds of arc.

If the striding level were perfect in construction, then
the reading obtained on reversal should be the same as

Zj Y-, In T o

that given previously. should = . Any

difference is due to an error in the striding level, and is
^qual to twice the striding level error. Thus the error

of the striding level itself- ** ~ r*-(l*-r*\

For example, if the left-hand readings of the bubble
of the striding level are 6-3 and 4-8, the corresponding
right-hand readings being 5-2 -and 6-8, we proceed as
follows : —

ASTRONOMICAL AND INSTRUMENTAL CORRECTIONS. 95

L. R.

6-3 5-2

4-8 6-8

11-1 12-0

11-1

4) 0-9

0-22

Therefore, if one division on the level corresponds to
14" inclination, the angle the axis makes with the hori-
zontal is 0-22 x 14= 3-1".

In this case the sum of the readings to the right is
greater than the sum of the readings to the left, and,
therefore, the right-hand end of the axis is the higher.
This would mean that the azimuth of a star (measured
from the North towards the right) would appear to be
greater than it really is, and the correction to be made
would consequently have to be subtracted. If the left-
hand end of the axis were the higher the correction to
-azimuth would have to be added.

If the preceding readings were taken with the striding
level on the transverse axis of a theodolite when a star
was being observed at an elevation of 42° 33' and the
azimuth reading was 127° 33' 10", the correction to be
made to azimuth would be 3. 1 x tan 42° 33' = 3-1 x -918
=2-8", and the corrected azimuth would be 127° 33' 7-2".

Allowance for Error of Alidade Level. — In most modern
theodolites intended for astronomical observations, no
level is attached to the telescope itself, but instead a deli-
cate level, known as the alidade level, is attached to the
vernier or microscope arms of the vertical circle, and the
circle turns with the telescope so that when the telescope
is horizontal the verniers are at zero.

With this form of instrument, when reading vertical
-angles, each reading should be repeated by changing
the face of the instrument, and to allow for any slight

96 ASTRONOMY -FOR SURVEYORS.

departure from true horizontality in the setting of the
theodolite, the alidade level should be read on each occa-
sion. In this case the readings of the two ends of the bubble
are commonly referred to as 0 and E, according as they
are at the object or eye end of the telescope.

The principle involved is exactly the same as that of
the striding level just described. The error in level will
be found by dividing the difference between the sums
of the readings of the object end and eye end by the total
number of readings, and then multiplying the result
by the angular value of one division of the scale of the
spirit level. If the readings of the object end are greater
than those of the eye end, then the zero line is pointing
slightly upwards, and the correction must be added on
to the observed altitude. If the readings of the eye end
are the greater, then the correction is to be subtracted .
So that

0— E

Correction to altitude — -\ — - X value of

1 division. number of readlng8

Thus, suppose that two observations are taken, one
with the face of the instrument to the left and the other
with face right, as follows :—

0. E.

F. L. 5 9

F. R. J7 _7

"12 ~i6

4 p4

Thus, if the angular value of one division on the level
is 14", it will follow that the altitude measured must be
reduced by this amount.

Clearly this correction applies to vertical angles only,
and does not affect the measurement of horizontal angles.

CHAPTER VIII.
THE DETERMINATION OF TRUE MERIDIAN.

THE determination by observation of a true North and
South line is a very important and common operation
for the surveyor, and there are many ways in which it
may be done. In practice, however, preference is given
to such methods as will allow a set of observations to be
taken so that instrumental errors may be eliminated,
some readings being taken with F.R. (face right) and
others with F.L. (face left), and also to such methods
as do not require too great an interval of time between
the observations. There is an objection to methods
which require stars to be sighted at an interval of several
hours, not only on the score of practical convenience,
but because the atmospheric refraction may have changed
considerably in the time that has elapsed. We shall
confine our attention to the principal methods in actual
use.

Referring Mark. — When determining the azimuth of
a star or other celestial object, it is necessary to have a
referring mark whose azimuth may be measured with
respect to that of the star, so that the true direction
may be found of a fixed reference object. It is commonly
indicated in field notes by the letters R.M. It is highly
desirable that there should be no need to refocus the
telescope after pointing it to a heavenly body and then
directing it to the referring mark, and this requires that
the referring mark should be where practicable about a
mile away. When stellar observations are being taken
the referring mark should be made to imitate the light

ASTRONOMY FOR SURVEYORS.

of a star as nearly as possible. This may be done with
a bull's eye lantern placed in a box or behind a screen,
through which a small circular hole is cut to admit the
light to the observer. The face of the screen may be
painted with stripes, so that it may be readily observable
in the day time. If the referring mark is not to appear
larger than a star in the field of view of the telescope,
the diameter of the hole must not be more than about
a third of an inch at a distance of one mile. Some observers
prefer a narrow vertical slit in the screen, and others use
a larger hole with two cross wires at right angles to each
other.

Fig. 25.

First Method — By Equal Altitudes of a Circumpolar Star.—
To mark out a true North and South line, we have
to determine the direction of the celestial pole, and the
simplest method is probably that of observing a circum-
polar star at equal altitudes. No calculations are necessary,
and no knowledge of the latitude, longitude, or local time
is required by the observer.

If the circle in Fig. 25 represents the circular path of a
star round the pole, the problem is to determine the

THE DETERMINATION OF TRUE MERIDIAN. 99

direction of the centre P of this circle. Suppose that the
star is observed at S, and then, keeping the angle of
elevation of the telescope unchanged, the observer waits
until he sees the star again at H at the same altitude.
Clearly the point L, midway between S and H, will be
vertically above the pole P, and all that the observer
has to do to get his true meridian is to bisect the angle
between S and H. Nothing could be simpler in principle,
but certain precautions are necessary to get accurate
results.

In the first place, when fixing either the points S or H,
we are really marking the point of intersection of the
horizontal line with the circle. Now, we can fix the
intersecting point of two lines most accurately when the
two lines are at right angles, and so the best position
for the line S H is when it passes somewhere near P.
As the star takes 24 sidereal hours to complete its circle
round the pole, this would mean that the second obser-
vation would be made about 12 hours after the first. This
would be often impossible and generally inconvenient.
It, on the other hand, the line S H is taken too near the
top of the circle, the star is moving so rapidly in a hori-
zontal direction that it is not possible to secure good
intersections.

Two simple observations at S and H, such as we have
just described, would not be sufficient to enable instru-
mental errors to be eliminated, and so in practice a set
of at least four observations are made, as illustrated in
Fig. 26. They will be made somewhat as follows : — Set
the instrument to zero and point to the R.M. Point to
the star in the position St, measuring the horizontal
angle between S, and the R.M., and noting also the
altitude of S,. Then change the face of the instrument
and point again to the star, which will by this time be
at S2. Again note horizontal angle and altitude. Keeping
the telescope clamped at the same vertical angle, unclamp

100

ASTRONOMY FOR SURVEYORS.

the upper plate and move the telescope round, waiting
until the star is again seen in the position S3. When
the star is got into the field of view of the telescope, the
upper plate is again clamped and the star followed by
means of the tangent screw until it again coincides with
the centre of the cross wires. Having read the horizontal
angle, the face of the instrument is again changed, the
altitude of the telescope is again set to the reading at S1?
and the star is again followed until at S4 it once more is

Fig. 26.

hi the centre of the field. Finally, the telescope is pointed
to the R.M.

The direction midway between S2 and S3 should, of
course, if there are no errors, coincide with that midway
between Sx and S4. This will not usually be the case,
but the mean of the two results is taken and instrumental
errors are largely eliminated.

If a, b, c, and d be the angles which S1? S2, S8, and S4
make with the R.M., then if the R.M. be outside the angle

THE DETERMINATION OF TRUE MERIDIAN 101

subtended by Sx S4 at the observer's eye, the angle tHat
the R.M. makes with the true meridian will be

+ c+d

If, on the other hand, the direction of the R.M. lies
between Sj and S4, the angle will be

a + b — (c + d)

The reason of this difference will be seen from Fig. 27,
where 0 P represents the true meridian bisecting the

angle between 0 Sj and O S4. If the referring mark is in
such a position as M1? outside the angle Sx 0 S4, the sum
of the angles Mx O Sj and Mj O S4 is double the angle
Mj O P. But if the referring mark is in such a position
as M2, within the angle S2 O S4, the difference of the
angles M2 O S, and M2 O S4 is double the angle M2 0 P.

The polar distances of the stars are not absolutely
constant, as the theory of the method assumes, but
undergo very slight changes during the year, which are
tabulated in the Nautical Almanac. In the course of
24 hours, however, the alteration never amounts to more

FOR SURVEYORS.

102

than a "small fraction of a second of arc, and, therefore,
need not be considered.

An unknown error may be introduced by changes in
the atmospheric refraction during the considerable in-
terval of time that must separate the first and second
sets of observations. The method will give results quite
sufficiently accurate, however, for the ordinary purposes
of the surveyor, it may be carried out without the use
of mathematical tables or Nautical Almanac, and it
involves no knowledge of the position of the observer.
Its great practical disadvantage is the length of time
over which the observations must extend, and to carry
them out the surveyor must be up for the greater part
of the night. Consequently other more convenient
methods are usually favoured by surveyors.

Second Method — By a Circumpolar Star at Elongation.—
In Fig. 28, let P be the celestial pole, Z the zenith
of the observer, W, S, and E the West, South, and
East points respectively on the circle of the horizon, or
the West, North, and East points according as the observer
is in the Southern or Northern Hemisphere. The small
circle with P as centre represents the path of a circum-
polar star A. The vertical plane passing through the
zenith of the observer and the star traces out the circle
Z A B on the celestial sphere. This will be the circle

THE DETERMINATION OF TRUE MERIDIAN. 103

swept out by the telescope of a theodolite when the
telescope, after being directed to the star, is turned in a
vertical plane about its transverse axis. As the star moves
from the position shown in the figure this vertical plane
will make a greater and greater angle with the plane of
the meridian Z P S until the star arrives at the position
H, where the vertical circle Z H K, swept out by the
telescope, is a tangent to the circular path of the star.
This is the point where the vertical plane containing the
star makes its greatest angle with the plane of the meridian.
At this point the star is said to be at elongation, and,
clearly, its motion being then vertical, it is in a favourable
position for observations upon its azimuth, because its
horizontal movement is so slight for some time before
and after it arrives at H. There will be a corresponding
point H' in the path of the star to the West of the celestial
pole, and the points H and H' are referred to as the
points of Eastern and Western elongation respectively.

It is clear from the figure that the points H and H'
will always be at a greater altitude than the celestial
pole P, but the smaller the circle of the star's path or the
greater the declination of the star, the more nearly will
the altitude of H and H' approach that of P.

Now, if a Nautical Almanac star is selected for obser-
vation, we shall know its declination, and the polar distance
P H is the complement of the declination. If, in addition,
we know the latitude of the place of observation, then,
in the right-angled spherical triangle Z P H, we shall
know P H and Z P, which is the complement of the
latitude. Hence, by Napier's rules, we can compute the
angle P Z H. We have

Sin P H = sin Z P sin P Z H

or Sin P Z H= cos declination x sec. latitude.

This calculation gives us the angle that the star at H
makes with the meridian. Hence, if we measure the

104 ASTRONOMY FOR SURVEYORS.

angles that the star at H makes with some referring
mark, the azimuth of the R.M. is determined.

The method so far indicated would require the direction
of the star to be measured at the exact moment of elonga-
tion. But we have set it down as a general principle
that at least two observations should be made, one with
F.L. and the other with F.R., and it bcomes important
to enquire what error in azimuth will be made if sufficient
time is taken to obtain two readings.

On making the necessary calculations, it will be found
that, for a place in latitude 30°, the azimuths of stars
at different polar distances will not alter by 5" after the
moment of elongation until the following times have
elapsed : —

Polar Distance Time after Moment of Elongation

of Star. before Azimuth changes by 5".

10°, 3 min. 33 sec.

15°, 3 min. 7 sec.

20°, 2 min. 35 sec.

30°, 2 min. 11 sec.

As there will be a corresponding and nearly equal
period before elongation, it follows that for a star whose
polar distance is 10° there will be a total time of about
7 minutes during which its motion is so nearly vertical
that the total change of azimuth in that period is not
more than 5". For a star whose polar distance is 30°,
the corresponding period is 4^ minutes.

If, then, the surveyor, as will commonly be the case
in ordinary work, is not seeking to determine the true
meridian nearer than within 20", it will be quite suffi-
ciently accurate to take two observations of the star, one
with F.L. and the other with F.R., not at the exact moment
of elongation, but one jus.t before and the other probably
just after elongation. The time required to read both
verniers, reverse face, and set the telescope again on the
star should not be more than three or four minutes, so
that there should be time to get both observations within

THE DETERMINATION OF TRUE MERIDIAN. 105

the period we have just calculated during which the
azimuth of the star does not alter by 5". The nearer the
star is to the pole the greater the length of time available
for the observations.

The average value of the angle that the star makes
with the meridian, as determined by two observations
in this way, is clearly always a little less than the angle
at elongation. In order to get the most accurate results
with this method, it is better not to use the formula for
the star at elongation at all, but to get a careful set of
four observations of the star near elongation, observing
the altitude of the star at each measurement. In Fig. 29,
let A represent the star moving in its circular path round

the pole P, Z the zenith, Z A B the vertical circle passing
through the zenith and the star. Then, in the triangle
Z P A, if the altitude of the star is measured, the values
of Z A (90°- the altitude) and ZP (the co-latitude) and
P A (the polar distance of the star) are known. If

P A= p= polar distance of star,

P Z= c= co-latitude,

Z A = z = zenith distance,
s=± (p+ c+z),

sin | P Z A

ysin (s-
-i

- z) sin (s — c)

sin z sin c

or log sin \ P Z A = J [log sin (s — z) -f log sin (s — c)
+ log cosec z -f log cosec c } .

106 ASTRONOMY FOR SURVEYORS.

Such a set of observations should be made in the
following order : — Point to R.M., point to star, reading-
altitude and horizontal angle, reverse face, and point
to star again. Turn back to R.M. and read angle. Then
another pair of observations are made in the same way.
The mean of the first two observations and the mean of
the second two are then used as the data for two separate
computations of the azimuth of the R.M. by means of
the formula we have just given. The average of the two
results, if the work is carefully done, will give a very
accurate determination. This is the method recom-
mended in the Hand Book of Instruction for Western
Australian Surveyors. An example is given a little further
on.

A more convenient method for reducing any number
of observations taken near to elongation is given at the
end of this chapter.

Calculation of the Time of Elongation. — In order to
prepare for these observations, it will generally be neces-
sary for the surveyor to work out beforehand the time
at which the star will elongate. In Fig. 28 the angle
Z P H measures, when turned into time, the sidereal
time that must elapse before the star at H comes on to
the meridian. But when the star is on the meridian the
sidereal time is given by the R.A. of the star. Thus the
sidereal time when the star is at H is= the R.A. of the
star— the hour angle of the star Z P H. This sidereal
time has then to be turned into mean time by the methods
we have previously discussed.

EXAMPLE. — To find the time of Eastern elongation of /? Centauri on April
IQth, 1914, at a place in 8. lot. 31°, longitude 135° E.

R.A. of /? Centauri, . " . .13 hrs. 57 min. 47-7 sec.
Dec. of p Centauri, . . .59° 57' 43-8" S.

We first of all find the time of culmination, the local sidereal time at
that instant beinir irivt-n by the R.A. of the star. Thus at culmination —

THE DETERMINATION OF TRUE MERIDIAN. 107

Local sidereal time,

Corresponding Greenwich sidereal

time, . ...

Sidereal time at G.M.N., April 10th,

13 hrs. 57 min. 47-7 sec.

4 hrs. 57 min. 47-7 sec.
Ihr. 11 min. 29-19 sec.

Interval in sidereal time after Green-
wich noon, ....
Interval in mean time after Greenwich

noon, .....
Local time corresponding to G.M.N.,

April 10th,

. • . Local mean time at culmination,
We have now to find the time from elongation to culmination, which
will be measured (Fig. 28) by the angle Z P H. .From the right-angled
triangle Z P H in that figure we have

cos Z P H = tan P H . cot Z P = cot dec. X tan lat.
cot dec. = cot 59° 57' 43-8", . . 9-7621015

tan lat. = tan 31C , 9-7787737

3 hrs. 46 min. 18-5 sec.
3 hrs. 45 min. 41-4 sec.

9 hrs. 0 min. 0 sec.
12 hrs. 45 min. 41-4 sec.

cos 69° 40' 10", . . . 9-5408752

.-. angle Z P H = 4 hrs. 38 min. 40-66 sec. sidereal time = 4 hrs. 37 min.
55 sec. mean time.

.-. time of Eastern elongation = 12 hrs. 45 min. 41-4 sec. — 4 hrs. 37 min.
55 sec. = 8 hrs. 7 min. 46-4 sec., April 10th.

Time of Western elongation = 12 hrs. 45 min. 41-4 sec. -f 4 hrs. 37 min.
55 sec. = 17 hrs. 23 min. 36-4 sec., April 10th, or 5 hrs. 23 min. 36-4 sec.
a.m. on April llth.

Azimuth, Altitude, and Hour-Angle at Elongation. — In
Fig. 30, if P denotes the celestial pole, Z the zenith of
the observer, and H a star at elongation. In the right-

108

ASTRONOMY FOR SURVEYORS.

angled triangle Z P H we have Z P = the co-latitude,
P H = the star's polar distance or the complement of
the declination, the angle Z P H = the hour angle of the
star, P Z H = the azimuth of the star if P is the North
celestial pole or the supplement of the azimuth if P is
the South celestial pole. Z H = the star's zenith distance
or the complement of the altitude. Hence we have the
following relations :—

cos Z P H = cot dec. x tan lat.

sin P Z H = cos dec. x sec lat.

sin altitude = cosec dec. x sin lat.

EXAMPLE OF OBSERVATION OF STAR AT ELONGATION FOR AZIMUTH.
Star— Canopus. Date— June 26th, 1914.

R.A .—6° 22' 01 -07". Place— Survey Office Tower, Adelaide.

Declination — 52° 38' 48". Latitude — 34° 55' 38".
R.M. taken— Obelisk on Mt. Lofty.
Computed approximate Standard Time of W. elongation, 4 hrs. 14 min.

OBSERVATIONS.

Object. Face.

Horizontal Circle.

Mean.

R.M. R
Star R
Star L
R.M. L

118° 09'

227° 44'
227° 43' 30"
11 8° 09' 00"

298° 09'
47° 44'
47° 44'
298° 09'

118° 09' 00"
:>27° 44' 00"
227 D 43' 45"
11 8° 09' 00"

Mean angle between star and R.M., 109° 34' 52". R.M. to East of Star.

CALCULATION.
Formula. — Sin A = cos declination x sec latitude.

log cos dec., 9-7829945

log sec lat., . . 10-0862497

log sin A, ."

A from South,

Azimuth of Star, .

Angle between R.M. and Star,

Bearing of R.M., .

9-8692442

47° 44'
227° 44'
109° 34' 52"
1 18° 09' 08"

THE DETERMINATION OF TRUE MERIDIAN. 109

io^.-l^

^•£

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(M 0

(M 00

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CO CO

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JO >O
Tf rti

50 2

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§10 »0 O
i— i <N (M

OO O5 CO CO

CO I> Oi O5

0000

00 GO 5<J «M

0000

<M (M 00 OC

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CO CO i— I •— i

^, ^ ^ ^

^ ^ ^ ^

*O CO •* CO

O O C\l (N

'cd

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CO !>• O5 Oi

r-H ^H O O

0000

00 00 <M <N

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CO i— I i— I (M

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0000

o o o o

00 GO <M (N

<N (M 00 GO

i— i Oi "H CO
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1-4 CO C<l i— I

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00 00

QC =

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110

ASTRONOMY FOR SURVEYORS.

COMPUTATION FOR AZIMUTH.

First Pair.

Second Pair.

Observed altitude,
Refraction, .
Corrected altitude,

45° 50' 37 -5"
55"
45° 49' 42 -5"

46° 16' 17-5"
54"
46° 15' 23-5"

Zenith distance z, . .
Co-latitude c, .
Polar distance p, . ,

44° 10' 17-5"
55° 04' 22"
37° 21' 13"

43° 44' 36-5"
55° 04' 22"
37° 21' 13"

2s, . . . '., , -

5, . . ' . ' . . .
S — C, > .

136° 35' 52 -5"
68° 17' 56"
13° 13' 34"

136° 10' 11-5"

68° 05' 06"
13° 00' 44"

S — Z, . . • .

24° 07' 38-5"

24° 20' 29-5"

L sin (s — z), : ;
L sin (.<? — c), . *
L cosec 2, . . . .
L cosec c? . . •"* .

9-6114752
9-3594456
10-1568864
10-0862497

9-6150814
9-3524891
10-1602513
10-0862497

L sin2 £ Z,
L sin fZ, . , . . .- .
*Z, . . .
Z, . ......
Azimuth of star, . -.
Angle to R.M., . -; .
Azimuth of R.M.,

19-2140569
9-6070284
23° 51' 58"
47° 43' 56"
132° 16' 04"
14° 07' 27-5"
118° 08' 36-5"

19-2140715
9-6070357
23° 52' 00"
47° 44' 00"
132° 16' 00"
14° 07' 15"
118° 08' 45"

Mean azimuth of R.M.,

11 8° 08' 41".

CALCULATION OF TIME OF ELONGATION.

G.S.T. of G.M.N., June 27th, 1914,
Allowance for longitude,

L.S.T. of L.M.N.,

R.A. of Canopus or L.S.T. of Culmina-
tion, . ...

Sidereal interval since L.M.N.,
Converted to mean solar time,
Correction to standard time,

Standard time of Culmination,

6 hrs. 19 min. 0-63 sees.
1 min. 31-06 sees.

6 hrs. 17 min. 29-57 sec.
6 hrs. 22 min. 01-07 sec.

4 min. 31-50 sec.
4 min. 30-75 sec.
15 min. 40 sec.

12 hrs. 20 min. 10-75 sec. p.m.

THE DETERMINATION OF TRUE MERIDIAN. Ill

cos hour angle at elongation = cot dec. X tan lat.
L cot dec. (52° 38' 47-25") =.9-8826803
L tan lat. (34° 55' 38") = 9-8440521

9-7267324

.-. hour angle = 57° 47' 28"

equivalent to . .3 hrs. 51 min. 9-87 sec. sidereal interval
or 3 hrs. 50 min. 32 sec. mean time interval

subtract from . . 12 hrs. 20 min. 10-75 sec.

giving . . .8 hrs. 29 min. 38-75 sec. a.m. as the standard time of

the Eastern elongation.

The Effect of an Error in the Latitude. — In the preceding
calculations we require to know the declination of the
star and the latitude of the place. The declination of
the star is given by the Nautical Almanac, but it is possible
that the latitude may not be known with the same degree
of precision. In Fig. 30, we have

sin Z cos I = cos d, . . ( 1 )

where I— latitude, d= declination, Z= angle P Z H.

Suppose that a small change y in the latitude produces
an alteration x in the azimuth Z. d remaining unaltered.

Then sin (Z+ x) cos (l-\- y) — cos d.

Expand each of these terms, remembering that x and
y are small, so that sin x, sin y may be replaced by x and
y respectively, and cos x, cos y by unity. We then get

(sin Z -+- x cos Z) (cos / — y sin /) = cos d.

Subtracting (1) from this equation and neglecting the
term involving the product of x and y

x cos Z cos I — y sin Z sin / = o,
or x — y tan / tan Z

sin Z

= y tan I

V(l-sin2Z)

cosd

= y tan I - from ( 1 )

V(cos2Z— cosad)

112

ASTRONOMY FOR SURVEYORS.

Thus x — o if l=o, and x = QC if / = d. I cannot be
greater than d, because if so Z P is les^ than P H (Fig. 30 ),
and the formulae would not apply. For such a star there
is no position of greatest elongation, as the azimuth of
the star during its revolution completes the circle of the
compass. If 1= d, the path of the star passes through
the zenith.

The following table gives the values of the error in
azimuth compared to the error in latitude, as calculated
by the preceding formula, for various values of / and d.

RATIO or ERROR ix AZIMUTH TO SMALL ERROR ix LATITUDE.

Declination of Star
Observed.

In Latitude 20°. In Latitude 80°.

In Latitude 40°.

60°

•22

•4

•7

70°

•14

•24

•4

80°

•06

•1

•19

In the cases tabulated an error in latitude of, sayr
5" will produce an error in azimuth of less than 5", the
tabulated ratios being all less than 1 . The error in azimuth
may, however, be much greater than the error in latitude,
if the star observed has a declination approaching the
value of the latitude.

In any given latitude, the error is least when the star
selected is nearest to the pole. From the formula, x= o
if d= 90°. This and other considerations, as we have
seen, all point to the desirableness of selecting a star for
observation as near to the celestial pole as possible.

Star Observations in Daylight. — It is often a very great
convenience to the surveyor to be able to make his obser-
vations for meridian in, the day time. The method that
we have just described of taking observations on a star
at or near elongation may be used perfectly well in day-
light, provided that a sufficiently bright star is selected.
Such work is done most easily in the late afternoon.

THE DETERMINATION OF TRUE MERIDIAN. 113

The following are suitable stars for such daylight obser-
vations in the Southern Hemisphere : —

a Argus (Canopus), a Eridani (Achernar),
a2 Centaur i, ft Centauri, a1 Crucis.

As these stars cannot be seen with the naked eye in
daylight, it is necessary to compute the position of the
one selected for observation before directing the tele-
scope to it. The time of elongation may be computed
by the method already discussed, and the azimuth and
altitude of the star at elongation determined by the
formulae given. When these calculations are made the
star may be readily picked up.

To select the most suitable star, compute roughly the
sidereal time when it is desired to make the observations.
A star must be selected which culminates some 4 or 5 hours
before or after this. That is to pay, the star chosen must
have a right ascension some 4 or 5 hours greater or less
than the computed sidereal time.

Fig. 31.

Third Method— Extra-Meridian Observations on Sun or Star. —

Suppose, in Fig. 31, that S denotes any heavenly body
which moves in a circle round the celestial pole P.
Let Z be the zenith of the observer. Then if the altitude
of S is observed at any instant, and if in addition we
know the latitude of the place and the declination of the
celestial body, then in the spherical triangle P Z S we

1U ASTEONOMY FOR SURVEYORS.

know the three sides S Z = z = 90° — altitude, PZ=c
= 90° — latitude, P S = p = 90° — declination. Conse -
quently, we can determine the angle Z which the vertical
plane through S makes with the true meridian.
If we write s = | (p-\- c-\- z), then

/si
= *J
v

sin (s- z) ,sin(«— c)
sin = *J . -T— ,

sin 2 . sm c

/sin s . sin (5 — p)

or cos

sm z . sin c

from which

log sin | Z = | {log sin (s — z) + log sin (s — c) + log
cosec s -f- log cosec c[ ,

and similarly for the second formula.

A more detailed discussion of these formulae is given
in the account of extra-meridian observations for time.

The method may be applied either to a star or to the
sun, but for the sun we require a little more information
than in the case of a star. To solve the spherical triangle
we must know both the latitude of the place and the
declination of the celestial object. In the case of a star
we can get the declination from the Nautical Almanac,
and as the declination changes very slightly through-
out the year, we only require to know the approximate
date of the observation in order to get the declination as
accurately as is necessary. But, with the sun, the declina-
tion changes very rapidly, and in the Nautical Almanac
its value is given at Greenwich mean noon for every day
in the year. In order to obtain the decimation at any
other instant, we must know the Greenwich time at the
moment in question, and this means that we must know
both the local mean time and the longitude. An error of
one minute in the time may produce an error of 1" in
the sun's declination. With the sun, therefore, the time
of observation must be noted as well as the altitude.

THE DETERMINATION OF TRUE MERIDIAN. 115

The method is also well suited for daylight observations
upon stars, as the very brightest stars are available for
this class of observation. Sirius (magnitude— 1-4) is a
very suitable star.

If the observation is made in the Northern Hemisphere
and S is to the East of the meridian, the angle P Z S is
the azimuth of the celestial body. If S is to the West
of the meridian, the azimuth == 360° — P Z S.

If the observer is in the Southern Hemisphere, then the
azimuth = 180°-PZS or 1SO°+PZS, according as S
is to the East or to the West of the meridian.

Extra Meridian Observations of a Star. — At least two
measurements of the altitude and the horizontal angle
made with the R.M. should be taken, one with the F.L.
and the other with F.R. Since the mean refraction for
objects at an altitude of 45° is 57", it is necessary to
correct for refraction in the measurement of the altitude.
As the proper correction for refraction is somewhat
uncertain for stars anywhere near the horizon, the star
selected for observation should have an altitude of at
least 30°. The order of procedure should be as follows : —

Point the telescope to the R.M.

Turn the upper part of the instrument round so as to
direct the telescope to the sta*r, reading both verniers
on the horizontal circle. Measure also the altitude of the
star.

Reverse the face of the instrument.

Again point telescope to star, measuring horizontal
angle and altitude.

Turn the upper part of the instrument, this time in the
reverse direction, until the telescope points to the R.M.

In the interval between the two pointings to the star
it will have moved considerably in altitude. If we average
the two altitudes and with the value so obtained solve
for Z by the formula given, the result will give us the
azimuth corresponding to this mean altitude, but that is

116 ASTRONOMY FOR SURVEYORS.

not exactly the same thing as the mean of the azimuths
in the two observed positions. Provided, however, that
the difference in altitude of the star at the two observa-
tions is not more than one or two degrees, the error thus
made is so slight that it is not worth considering.

When the observations are to be made upon a bright
star in the day time, it will be necessary, first of all, to
compute the azimuth and altitude of the star for the time
of the first observation in the manner explained and
illustrated in Chapter VI. The azimuth and altitude
5 or 10 minutes later may then be deduced, as shown in
the same chapter.

Extra Meridian Observations upon the Sun. — The sun,
being an object of large size in the field of view of the
telescope, cannot be observed in the same way as the

234-

Fig. 32.

stars. The observer must sight to its edge, and in this
case, wh#re both horizontal angle and altitude are to be
measured, it may be sighted in any one of the four quad-
rants formed by the cross wires of the telescope. The
four different positions in which it may be observed are
shown in Fig. 32, the two cross wires at right angles
being brought by means of the tangent screws so as to just
touch the sun's edge in each case. The centre of the sun's
disc is the point considered in all our computations, and
this then is the point whose position we seek to determine.
Clearly, the centre of the cross wires is midway between
the centres of the sun discs in positions 1 and 3, so that
the mean of the readings in these two positions should
give us the altitude and azimuth of the sun's centre.

THE DETERMINATION OF TRUE MERIDIAN. 117

Similarly the mean of the readings in positions 2 and 4
will give the position of the sun's centre. A complete
set of observations will consist of four observations of
the sun in the four positions illustrated. They should be
made as follows : —

Take reading of R.M. and clamp horizontal plate.

Turn to the sun and observe altitude and horizontal
reading with the sun in quadrant 1 of the cross-wire
system.

Then, as quickly as possible, by means of the two
tangent screws, bring the sun into quadrant 3 of the
cross wires, and again read horizontal angles and altitude.

Turn back to the R.M.

Reverse the face of the instrument and take two more
observations in precisely the same way, but this time
with the sun in quadrants 2 and 4.

Be careful to note the time of each observation.

During the whole time occupied by the four observa-
tions the sun's position will have changed too much for
accurate results to be obtained by averaging the measured
altitudes and times of the four observations. There
should, however, be very little time lost between the
first two readings, with the sun in quadrants 1 and 3,
and the measured altitudes and times of these two may
be averaged together and a computation made for the
corresponding azimuth of the referring mark. Similarly,
another computation is made, by averaging the readings
with the sun in quadrants 2 and 4, from which the azimuth
of the referring mark is again determined. Thus we
obtain two computed azimuths, one with each face of
the instrument, and the average of the two is taken.

The two succeeding observations made without change
of face in quadrants 1 and 3 or quadrants 2 and 4 are
sometimes a little simplified by what is known as the
" run through " method. In this method the observer,
after making the first observation, leaves the telescope

118 ASTRONOMY FOR SURVEYORS.

clamped in vertical arc, and makes the second observation
when the sun has just crossed the horizontal wire by moving
the vertical wire to the correct position, with the aid of
the tangent screw attached to the horizontal circle. The
necessity of recording a second set of vertical angles is
thus avoided. The objection to this method is that the
two observations cannot be made in such quick succession
as is possible by the method outlined above, and conse-
quently the error made by taking the average of the two
observations is greater.

Very commonly only two observations of the sun are

made, and in that case the best procedure is as follows :—

1. Observe the R.M., say, with face L. 2. Observe the

sun in, say, quadrant 1 with face L. 3. Reverse face and

1234

Fig. 32a.

observe the sun again as quickly as possible with face
R. in quadrant 3. 4. Observe R.M. again with face R,
The average of the two observations is then taken as the
basis of a single computation.

Should the telescope have its cross wires of the form
shown in Fig. 32a, the observations will be precisely the
same, but the various positions of the sun's image will be
as illustrated.

For good work the altitude readings should always be
corrected by means of the alidade level, reading the E.
and O. ends at each observation.

Computation ol Sun's Declination from Nautical Almanac
Data. — In the Nautical Almanac the sun's declination is
given for both mean and apparent noon at Green wich,

THE DETERMINATION OF TRUE MERIDIAN. 119

for every day of the year, and also its rate of variation in
one hour at Greenwich noon. If the declination is required
at, say, 8 hours after Greenwich noon, it will not be accu-
rately found by multiplying the hourly variation by 8
and adding or subtracting the result to the value of the
decimation at Greenwich noon, because the hourly
variation itself is not constant, but changes from hour
to hour. The proper plan is to find the mean value of
the hourly variation over the interval in question, which
in this case will be the value at the middle of the interval
— i.e., 4 hours after noon.

EXAMPLE. — Required the value of the sun's declination at 9 hrs. 20 min.
a.m. on August 2nd, 1914, the time being South Australian standard, that of
the meridian 9 hrs. 30 min. E.

Corresponding astronomical time, . August 1st, 21 hrs. 20 min.

Corresponding Greenwich time, . August 1st, 11 hrs. 50 min. p.m.

Hourly variation at G.M.N. on August 2nd, 38-05"
Hourly variation at G.M.N. on August 1st, 37-32"

0-73"

The half of 11 hrs. 50 min. is very nearly 6 hrs. Therefore, the average
hourly variation is

37.32+^=37-5

11-83 hrs. x 37-5 - 443-6" - 7' 23-6".
Sun's declination at G.M.N., August 1st, 18° 10' 50-4" N.,
and it is decreasing at this time of the year.

.•. Sun's declination at given time, . 18° 03' 26-8" N.

Corrections to Sun Observations.— -We have already seen
in Chapter VII. that the sun is one of those bodies the
observed altitude of which must be corrected for parallax.
It must also be corrected for Refraction, as shown in the
same chapter.

Either from want of time or through the intervention
of clouds the surveyor may be unable to complete the
series of four observations, but any single observation
will enable him to determine the position of the sun's

120

ASTRONOMY FOR SURVEYORS.

centre by making proper allowance for the sun's semi-
diameter, the value of which is tabulated in the Nautical
Almanac.

There is no difficulty with regard to the determination
of the altitude of the sun's centre from one observation,
as the semi-diameter has simply to be added on or sub-
tracted as the case may be. If, for instance, with a re-
versing telescope the sun is observed in quadrant 1, it
will mean that we are actually sighting the upper edge
of the sun, and the measured altitude will have to be
reduced by the value of the semi-diameter given in the
Nautical Almanac.

But with the observations for azimuth the matter is
not quite so simple. Thus, in Fig. 33, if C denotes the
centre of the sun's disc, Z the zenith, Z C A the vertical
trace on the celestial sphere passing through C and the
zenith, Z P B the vertical plane just touching the edge
of the sun's disc, then the error in azimuth made by
sighting the edge instead of the centre of the sun's disc
is the angle C Z P. But in the right-angled triangle
Z C P we have

sin C P- sin C Z . sin C Z P.

Now C P is an angle of about 15 minutes, and
its circular measure differs from its sine by 1 only

THE DETERMINATION OF TRUE MERIDIAN. 121

in the seventh place of decimals. Consequently, we
may write

CP=sinCZx CZP

CZP=CPx cosecCZ,
or correction in azimuth

= semi-dia. x sec. altitude sun's centre.

The Effect of an Error in Latitude upon the Calculated
Azimuth. — Referring to Fig. 31, we shall determine the
effect of an error in latitude if, in the spherical triangle
P Z S, we investigate the effect upon the angle Z of a
small change in c, the sides p and z remaining constant.

Let x be the change produced in Z by a small alteration
y in c. Then

cos p = cos c cos z + sin c . sin z cos Z (formula (2)

Chap. I.)

and cos p= cos (c-\- y) cos z-\- sin (c+ y] sin z cos (Z+ x).

Subtracting these two equations, writing x and y in
place of sin x and sin y, and unity in place of cos x and
cos y, we get

O = cos z . y . sin c + sin z sin c cob Z — sin z (sin c

+ y cos c) (cos Z — x . sin Z)
= cos z . y sin c — y sin z . cos c cos Z
-f x . sin z . sin c . sin Z,

neglecting the term involving the product of x and y.

—cos z . sin c + sin z cos c . cos Z

.-. # = — — -: : : — = - y

sin z sin c sin Z

. v (by formula (3) of Chap. I.)
sin c sin Z

- cot P
sin c

P is, of course, the hour angle, and we thus have a simple

122

ASTRONOMY FOR SURVEYORS.

formula for computing the error in azimuth produced
by a given error in latitude at any given time of the day.
Clearly, when P is very small — that is to say, at times
near to noon — cot P is very great, and the error produced
by a defective knowledge of the latitude is much increased.
In Fig. 34 a curve is drawn showing the error in azimuth
produced by an error of one second in the latitude, at
different hours of the day in latitude 40°. It is really a
curve of tangents, and it will be seen that the error is very
much greater at or near noon than at any other time. The

Fig. 34. — Error in Azimuth for Extra Meridian Observation of the Sun,
corresponding to error of one second in Latitude, at different hour&
of the day in Latitude 40°.

error is least at 6 a.m. or 6 p.m. With increase in the
latitude of the place of "observation the error would be
greater still, becoming very great for latitudes near the pole.
The Effect of an Error in the Sun's Declination upon the
Calculated Azimuth. — If a slight alteration y is made in
the value of p (Fig. 31), c and z remaining constant, then

THE DETERMINATION OF TRUE MERIDIAN. 123

it may be shown in a similar manner to that of the work
just preceding that

x = cosec c cosec P . y,

where x is the corresponding change made in the azimuth
Z. The establishment of this formula we will leave as
an exercise for the student.

In Fig. 35 a curve is drawn showing the error in azimuth
produced by an error of 1 second in the declination at
different hours of the day at a place in latitude 40°.

p.m.Gh. 54321

Fig. 35. — Error in Azimuth for Extra Meridian Observation of the Sunr
corresponding to error of one second in Declination, at different hours
of the day in Latitude 40°.

Again the error is very great near mid-day, and is least
at 6 a.m. and at 6 p.m. As in the previous case, with
increase in latitude of the place of observation the error
also increases, becoming so great in latitudes near the
pole that the method would be quite unreliable in arctic
or antarctic regions.

124 ASTKONOMY FOR SURVEYORS.

It will be noticed that the two errors we have just
discussed are of opposite signs, so that, if the declination
and latitude are both too large, the errors tend to neutralise
one another.

The Effect of an Error in the Longitude of the Place of
Observation. — An error in longitude will produce an error
in the computed Greenwich time at the instant of obser-
vation, and this in turn will produce an error in the
calculated declination. An error of 1° in longitude will
produce an error of 4 minutes in time. Now the rate of
change of the sun's declination varies at different seasons
of the year, but its maximum rate of change is less than
1 minute of arc per hour. Thus an error of 1° in longitude,
or 4 minutes in time, will produce an error in the declina-
tion that is always less than 4 seconds. As we have just
seen, the resulting error in azimuth is never less than the
error in declination, but if the observation is not made
within two hours on either side of noon, the azimuth
^rror is not much more than the declination error. It
will thus seldom happen that the longitude is not known
-approximately enough for the purposes of the surveyor.

The Effect of an Error in the Measured Altitude.— Referring
again to Fig. 31, we have in the spherical triangle S P Z
cos p = cos c cos z + sin c . sin z cos Z.

Let x denote the small change in Z produced by a small
change y in z, p and c remaining constant. Then

cos p = cos c . cos (y -j- z) -f- sin c . sin (y + z) cos (x-\- Z).

Subtracting and simplifying these equations, regarding
x and y as small quantities, we finally arrive at the result
x=— cot S cosec z . y.

Thus x will be infinitely great when S = o or 180°, which
is the case when the sun is on the meridian. And, again,
we arrive at the result that the resulting error in azimuth
is very great if the observation is made near noon, but
is small if S is anywhere near 90°.

THE DETERMINATION OF TRUE MERIDIAN. 125

-d l4 M

9^ CO

•

O5 <M rj
<M C0-|

» o 3

a a*

111

b »o

i— 1 r-H

§5 3

CO r-H

0 0

O 0

b b

o r— i

§3 ^

S S

0 0

^*J

0 0

b b

<N (N

co o

<N Tj*

CO ^

10 ^H

0 0

<N (M .

i— i (M

5 5

<M <M

5<J b

CO O

0000

CO IO

O CO CO O
CO O O CO
CO <N <N CO

CO O O CO

o b

(N (N

(M CO

CO O

0000

O CO CO O
00 C^ G^ CO

CO <N <M CO

'§ 1

<M CO

CO 0

0000

O CO CO O
CO C^ C^ CO

GO <N G^ 00

ft?

a c

H .2

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^ 1

S CO l>

F-H r-H
O O

1 I

X C* <N

CO CO

m.f

IE*
O

^ Ol IO ^

« 2! '° S

126

ASTKONOMY FOR SURVEYORS.

The Best Time for Extra-Meridian Observations. — The
preceding discussions all point to the desirableness of
making the observations upon the sun as far away from
noon as possible. But if we observe it when too low down

COMPUTATION.

F.».

F.L.

Standard time of obser-
vation, June 25th, .
Longitude East,

2 hrs. 56 min. 45 sec.
9 hrs. 30 min.

3 hrs 01 min. 31 sec.
9 hrs. 30 min.

Corresponding G.M.T.,
June 24th,

17 hrs. 26 min. 45 sec.

17 hrs. 31 min. 31 sec

Sun's declination at
G.M.N., .
Variation since G.M.N.,

23° 26' 09 -4"
46-4"

23° 26' 09-4"
46-6"

Sun's declination when
observed,
Observed altitude,
Refraction and parallax,

23° 25' 23"
20° 35' 42-5"

2' 22"

23° 25' 22-8"
20° 02' 00"

2' 27"

Corrected altitude,

20° 33' 20-5"

19° 59' 33"

Zenith distance = z, .
Sun's polar distance = p,
Co-latitude = c, .

69° 26' 39-5"
113° 25' 23"
55° 04' 21 -5"

70° 00' 27"
113° 25' 22-8"
55° 04' 21-5"

2s, .

237° 56' 24"
118° 58' 12"

238° 30' 11-3"
119° 15' 05-6"

s-p,
L sin s, ...
L sin (s — p),
L cosec z, .
L cosec c, .

5° 32' 49"
9-9419452
8-9852526
10-0285705
10-0862505

5° 49' 42-8"
9-9407569
9-0066890
10-0269935
10-0862505

L cos2 4 Z, .
L cos £ Z, .
*Z, . . . .

Z (from South), .

19-0420188
9-5210094
70° 36' 57"
141° 13' 54"

19-0606899
9-5303449
70° 10' 38"
140° 21' 16"

Bearing of sun, .
Angle between sun and
R.M.,

-321° 13' 54"
156° 55' 40"

320° 21' 16"
157° 48' 30"

Azimuth of R.M.,

118° 09' 34"

118° 09' 46"

THE DETEKMINATION OF TRUE MERIDIAN. 127

in the heavens, the refraction becomes a very uncertain
quantity, and consequently it is impossible to measure
the altitude with precision. For this reason it is generally
considered inadvisable to make the observation with the
sun at a lower altitude than about 15°. With this limi-
tation it is desirable, in order to minimise the effects of
errors in altitude, latitude, and declination, to make the
observation as far from noon as possible. So that if the
readings are made in the morning, they should be made
as soon as possible after the sun has reached an altitude
of 15°. Similar remarks will apply to the stars, which
should be observed as far away from the meridian as
possible, so long as they are at an altitude of at least
15° above the horizon.

Fourth Method — Time Observations upon a Close Circum-
polar Star. — The method about to be described is the one
chiefly adopted on geodetic surveys where the highest
attainable degree of accuracy is desired. The observa-
tions consist in measuring a series of angles between a
close circumpolar star and the R.M., noting the time at
which each pointing is made to the star. No altitudes
need be measured, and as the time may be measured with
sufficient precision by means of a chronometer, the
method is simple, as well as capable of great accuracy.
In the Northern Hemisphere the star a Ursse Minoris
(Polaris) is a very convenient one for the purpose. Being
a star of the second magnitude, it can be readily found,
and it is within about 1° 10' of the N. Pole. \ Ursse
Minoris is within 1° of the Pole, but is a much fainter
star, being of magnitude 6- 6. Other suitable Northern
circumpolar stars are 51 Cephei (Mag. 5*2) and S Ursae
Minoris (Mag. 4*4). In the Southern Hemisphere, un-
fortunately, there are no stars near the pole sufficiently
bright to be readily picked out without first of all cal-
culating their positions. The best star for the purpose
is ff Octantis, which is within 46' of the S. Pole. It is,

128

ASTRONOMY FOR SURVEYORS.

however, of magnitude 5-5, and in order to pick up the
star it is necessary to know beforehand the approximate
bearing of the R.M. This may be found from a daylight
observation by one of the methods previously described.

In Fig. 36, let P be the celestial pole around which
circulates in a small circle the circumpolar star S. Let
Z be the zenith. Then in the spherical triangle Z P S,
Z P = c = co-latitude, P S = p = polar distance of star,
Z P S = t = hour angle of star. P Z S = Z = azimuth
angle of star.

Fig. 36.

From formula (3) in Chapter I.

cot p sin c = cot Z sin t -f cos c cos t,

cot p sin c — cos c cos t
cotZ=-

sin t

sin (c — x) cot £
sin x

where tan x = tan p cos £.

The hour angle t, in time, is found by taking the differ-
ence between the R.A. of the star, which is the sidereal
time when the star is on the meridian, and the sidereal
.time at the moment of observation. To determine this
we must know both the local mean time and the longitude
of the place. Thus, we require to know the R.A. and
declination of the star and also the latitude and longitude
of the place of observation.

THE DETERMINATION OF TRUE MERIDIAN. 129

For the most accurate work the striding level should be
used to determine the error in the measured azimuth of
the star owing to any defect in the levelling of the trans-
verse axis of the telescope. This will produce an appreci-
able effect upon the azimuth of the star owing to its alti-
tude, but as the R.M. will usually be near the horizon,
it will not as a rule be necessary to apply any correction
on this account to the reading taken to it. If, however,
the R.M. should be at a considerable altitude, it would
be necessary to read the striding level both when the
telescope is pointed to the star and when it is directed
to the R.M.

The series of observations necessary may be arranged
in several different ways. The following is the programme
recommended by the U.S. Coast and Geodetic Survey :—

1. Point twice upon the R.M. and read the verniers
of the horizontal circle at each pointing, the instrument
being F.L.

2. Read twice on the star with F.L., noting at each
pointing the exact time, the reading of each end of the
striding level, and the readings of the horizontal circle.

3. Read twice on the star with F.R., the instrument
being reversed, noting the time and bubble readings as
before.

4. Read twice upon the R.M. with F.R.

According to this programme the striding level is left
with the same ends on the same pivots throughout the
observations.

The programme suggested in the handbook of instruc-
tions for Western Australian Surveyors is as follows : —

1 . Set the instrument to zero, point to R.M., and read
the circle.

2. Intersect star and take the time.

3. Read the striding level and reverse it.

4. Read the circle.

5. Intersect star again and take the time.

130 ASTRONOMY FOR SURVEYORS.

6. Read the striding level.

7. Read the circle.

8. Point to R.M. and read the circle.

In turning back to R.M. the instrument is moved in
the opposite direction. The instrument is now reversed,
the setting on the R.M. increased by 22° 30', and the
operation repeated until angles have been read all round
the circle.

A series of observations having been taken by one of
these systems, the hour angle of the star and the corre-
sponding azimuth will, of course, be different for each
pointing. Each separate observation will give us the
azimuth of the R.M., and we wish to get the mean or
average of these determinations. We may compute the
azimuth of the star for each pointing separately by means
of formula (1), deducing from each computation the
azimuth of the R.M., and then take the average of the
different results. This is the simplest procedure, involving
no mathematical difficulties, and when only a few obser-
vations have been taken this is the best plan to adopt.
But when there are a number of observations the calcu-
lations may be lessened by computing the azimuth corre-
sponding to the mean of the several hour angles. This
would not be the same as the mean of the different azi-
muths, but the latter may be derived from the former
by applying a correction known as the " curvature correc-
tion/' In the case of a close circumpolar star, and a
series of observations not extending over about half an
hour, the curvature correction is given by the formula

12 sin«j(T-T0)

Correction = tan A - £— — -,

n sm 1

where A = computed azimuth of star at the mean hour

angle of n pointings,
T0 ~ mean of the n hour angles.
T = any one of the separate hour angles.

THE DETERMINATION OF TRUE MERIDIAN. 131

The establishment of this correction is rather beyond
the mathematical scope of this work.

The true mean azimuth always lies nearer the meridian
than the azimuth corresponding to the mean hour angle.

The expression — — , which is usually

sin 1

denoted by m, has to be evaluated for each observation,
and as the same form also enters into the computation
of circum-meridian observations for latitude, tables have
been computed, available in various works, such as
Chauvenet's Astronomy and Trigonometrical Surveying
by Major Close, in which the values are tabulated for
different values of T— T0. The use of such tables greatly
facilitates the computation, as the curvature correction
is then found by adding the different values taken from
the tables, dividing the sum by n and multiplying by
tan A. A table giving the values of m at intervals of
10 seconds of time up to 19 minutes is given at the end
of Chapter IX.

Cireum-Elongation Observation for Azimuth.

The following account is extracted from a paper by the
author published in the Transactions of the Royal Society
of South Australia, vol. xxxix., 1915. The mathematics
involved is rather more advanced than that in the rest
of this work, but the method is of sufficient importance to
make it desirable to insert it : —

On account of its convenience and comparative sim-
plicity, the observation of a circumpolar star at elongation
is, amongst surveyors, the favourite star observation for
the determination of a true azimuth. The great dis-
advantage of the method is that only one observation
can be made with the star actually at elongation, and
there is thus no opportunity to eliminate instrumental
errors in the same way as may be done, when a series of

132 ASTRONOMY FOR SURVEYORS.

observations of the same star are made, by taking half
the readings with the instrument reversed. As a rule
the motion of the star in azimuth is so slow, when near to
elongation, that with an ordinary transit theodolite two
observations can be made and treated as though the star
were actually at elongation without introducing an error
sufficient to be measured by the instrument. But a much
higher degree of accuracy is possible with the method
if a series of half a dozen observations are made on each
side of elongation, and the object of the present paper is
to discuss the convenient reduction of such a series of
observations. For the reduction of a similar set of obser-
vations made upon a close circumpolar star there is a well-
known method that is particularly applicable to the Pole
star of the Northern Hemisphere. Unfortunately in the
Southern Hemisphere the close circumpolar stars are
very faint and not easy to work with, a Octantis has a
polar distance between 46' and 47', but its magnitude
is 5J, so that it is not readily picked out by the surveyor.
The bright southern stars that are most convenient for
the determination have commonly a polar distance of
about 30°, and to these the formula for close circumpolar
stars cannot always be applied without introducing
appreciable error.

Two methods are possible for a series of observations
made before and after elongation. We may read the
verniers of the horizontal circle and note the time at each
observation, or we may read the horizontal circle and also
the altitude of the star at each observation. The former
method is preferable, provided that the surveyor has the
correct local time, as errors due to a defective knowledge
of atmospheric refraction are not then introduced. The
latter method, however, involves no knowledge of the
time, and is much more convenient when the observations
have to be carried out single-handed. In both cases the
azimuth of the star at each observation is corrected by

THE DETERMINATION OF TRUE MERIDIAN. 133

the appropriate formula to give the azimuth of the star
at elongation, so that practically we obtain a series
of observations at elongation instead of only one.

Notation.

The following abbreviations will be used throughout : —

z denotes the zenith distance of the star in any position.
p ,, polar distance of the star.

A ,, horizontal angle between star and pole.

I „ latitude of place of observation.

c „ co-latitude of place of observation.

h „ hour angle of the star in angular measure.

t ,, value of hour angle expressed in sidereal

time.

z0, A0, h0, and tQ denote the values of z, A, h, and t
respectively when the star is at elongation.

First Method — Horizontal Angle and Time being Noted at
each Observation. — In the spherical triangle having the star,
the celestial pole, and the zenith as its angular points we
have the following fundamental relations :—

cos A sin z = cos p sin c — cos c sin p cos h, . (1)
sin A sin z — sin p sin h, (2)

and from the corresponding right-angled triangle when
the star is at elongation

sin p cos p cos h0

8in-40= — = . . . (3)

sin c cos c

cos A 0-= cos p sin h0. . . . , (4)
(1)X (3)-(2)x (4) gives

sin z sin (A^ — A] = cos p sin p 2 sin2 J (A0 — h). . (5)

134 ASTRONOMY FOR SURVEYORS.

This is an exact equation, but is unsuitable as it stands
for use in reduction of observations.

sin p sin A

Putting - - = •" -, (5) may be written

sin z sin h

sin (A0— A) 2 sin2 J (h0 - h)

-—- - = cos p - — ; — —
sin A sin h

2sin2i(ft0-ft)
or, writing y = cos p - — —

sin A0 cot A — cos A0= y.

A0 is constant, and, therefore, A may be regarded as a
function of y.

Differentiating, we have

1 dA

— smA0 ~ ~~— - 1,
sin2 A dy

d2 A dA

and — sin A0 - = 2 sin A cos A — -.

dy2 dy

Therefore, when y = o

—= — sin A0 and - - = sin 2 A0,
dy dy2

and consequently, by Taylor's Theorem

. cos p 2 sin2 J (hQ — h)

A = A 0 Sin A o - ; r— ; — — -

sin h sin 1
cos2 p 2 sin4 J (ft0 — h)

+ sin 2

sin2/? sin

provided that A0— A is measured in seconds of arc.

This is a convenient converging series for the deter-
mination of the difference between A and A0, in which
the terms diminish so rapidly that in all ordinary work
it is not necessary to take into account any term except
the first. Thus, if the observations are made at a place
in latitude 30°, on a star with a polar distance of 30°,

THE DETERMINATION OF TRUE MERIDIAN. 135

and are continued for fifteen minutes of time on each side
of elongation, the extreme value of h — h0 = 3° 45'. The
corresponding value of the first term in the series then
works out at 229", or 3' 49", and that of the second term
at less than \" '. If t— t0= 30 minutes, or h— /?0 = 7° 30',
then under the same conditions the first term= 902"
and the second term only 5|". With the same polar
distance and in the same latitude, the limiting value for
t— tn, in order that the second term may not be greater
than 1", is about 19 minutes. On repeating the calcu-
lations for a place in latitude 20°, and again for a place in
latitude 40°, it is found that in neither case does the
limiting value of t — t0 differ by more than a minute from
the value previously found if the second term in the series
is to be less than 1".

It thus appears that, even if the mathematical reduction
of each single observation is to be correct within 1" of
arc, it is sufficient to use only the first term of the series
if the observations extend over a period of about 19
minutes on each side of the elongation. The average
of the whole series may be correct within this limit, even
if the time extends over a considerably longer period,
because the error in reduction will exceed 1" only in the
case of the extreme observations.

A further considerable simplification would be made in
the reduction if it were possible to treat the denominator
as constant and write sin h0 instead of sin h. With any
single observation the error made, if this is done, may
be considerable. For instance, at a place in latitude 30°,
if p= 30°, for an observation made 15 minutes before
elongation, the difference made in the value of the second
term, when sin h0 is written in the denominator instead
of sin h, is about 5", whilst for an observation made
30 minutes before elongation the difference is about 35".
But, if we have a series of fairly well-balanced observa-
tions made both before and after elongation, the values

136 ASTRONOMY FOR SURVEYORS.

of h range fairly evenly on each side of k0, and on averaging
up the set there will be very little difference whether we
use h or h0, the difference being generally of the order
of 1". So that in such a case it is usually quite sufficient
for the surveyor to use #0 instead of h. We may then
make a further slight simplification by putting
Min A 0 cos p

sin h0

= tan A0 cos2 p.

Practical Computation. — We therefore conclude that,
for the ordinary work of the surveyor, a series of well-
balanced observations extending to about half an hour
on each side of elongation on any circumpolar star may
be reduced to a series of observations at elongation by
the formula

Att-A = t1mAn^p2^4^~^,. . (6)

sin 1

in which A0— A is given in seconds of arc.

If, however, only one or two observations are to be
reduced, as may be the case if the star at elongation has
been obscured by clouds, or the observations are badly
balanced and have been made mostly on one side of
elongation, or if the greatest possible degree of accuracy
is required in the computations, the formula used should be

A A- sin A C°S P_l™lii*»- h) m

sin h sin 1"

This form may be obtained directly from (5) by con-
sidering AQ — A as a small angle so that the sine may be
written equal to its circular measure.

If it is required to make the computation within I",
then, for observations more than 18 minutes from elonga-
tion, the value of A0— A given by formula (7) should be
corrected by being decreased by the amount

Sm2VOSX2^.4(ArA). (8)

sm2 h sin 1

THE DETERMINATION OF TRUE MERIDIAN. 137

As the expression has to be evaluated

sm 1

in the reduction of circum-meridian observations for lati-
tude, tables of the value of the expression and its logarithm
have been prepared, and are available in Chauvenet's
Astronomy, Close's Astronomical Surveying, and other
works. An abbreviated table is given at the end of

Chap. IX. Similar tables for are also

sin 1

available. The computation by any one of these formulae
is much facilitated by the use of these tables. Five-
figure logs are sufficient.

Writing tan A0 cos* p = B, m = .2ain'* (V^*).

Sill 1

(6) becomes

AQ — A = B m, where B is a constant.

Thus for each observation we get ^40— A -f- B m, and,
averaging the whole series,

Mean value of A0 = mean value of A + B x mean
value of m.

Therefore, mean angle between R.M. and star at elonga-
tion = mean observed angle between R.M. and star
i B x mean value of m.

EXAMPLE. — In the following example the method is
applied to the reduction of a series of observations taken
by Mr. Calder, surveyor, upon Canopus near elongation : —

Star observed — Canopus.
Place — Rendelsham, South Australia.
Right Ascension — 6 hrs. 22 min. 06 sec.
Latitude— 37° 32' 40" S.
Declination— 52° 38' 43" S.
Longitude — 9 hrs. 20 min. 40 sec. E.
Date— December 9th, 1914.
Standard Meridian — 9 hrs. 30 min. E

138

ASTRONOMY FOR SURVEYORS.

COMPUTED VALUES.

Standard time at elongation — 9 hrs. 45 min. 32 sec. p.m.

A0 = 49° 55' 44"
h0 = 54° 04' 50"

Interval of Corres-

Face.

Object.

Mean Vernier
Readings on
Horizontal Circle.

Standard Time
of
Observation.

Mean Time ponding
between Interval in
Observation Sidereal
and Time.

Elongation.

R.M.

360°

H. M. S.

min. sec. min. sec.

Star

83° 16' 00"

9 32 44

12 48 ! 12 50

Star

83° 15' 15"

9 34 37

10 55 10 57

R.M.

360°

Star

83° 13' 45"

9 38 25

7 07 7 08

Star

83° 13' 00"

9 40 15

5 17 5 18

R.M.

360°

Star

83° 12' 15"

9 43 05

2 27 2 27

Star

83° 12' 45"

9 45 11

21 21

R.M.

360°

Star

83° 12' 15"

9 48 40

3 08 3 09

Star

83° 13' 15"

9 50 55

5 23 5 24

R.M.

360°

Star

83° 16' 45"

9 58 17

12 45 12 47

Star

83° 18' 15"

10 01 00

15 28 15 31

R.M.

360°

Mean observed angle between star and R.M., 83° 14' 21".

Solving by means of (6), we obtain from the tables :—

min.

sec.

10)

323-3'

235-4'

99-9'

55-r

11-8'

0-2'

19-5'

57-2'

320-8'

472-6'

1.595-8

THE DETERMINATION OF TRUE MERIDIAN. 139

Mean value of m

log tan A0 = 10-07509

log cos2 p = 9-80062

log 159-6 = 2-20303

log 120 = 2-07874
.-. Bm = 120" = 2'

.•. Mean value of angle between R.M. and star at elongation
= 83° 14' 21" — 2' 0" = 83° 12' 21"

The computation by means of the more accurate
formula (7) is rather longer. In this case we write

2 sin1 1 &— A)

B = sin AQ cos p and m = ~~ —- — ,

sin h sin 1

and work on the same lines as before. To illustrate the
method the computation in this case is also worked out
as follows : —

log m

= differ-

t.-t. h*-h. .

10g2ShSn^-

log sin h.

ence of
two

HI.

preceding

columns.

min. sec.
12 50 3° 12' 30"

57° 17' 20'

12-50960

9-92501

2-58459

394-2

10 57 2° 44' 15" ' 56° 49' 05'

12-37178

9-92269

2-44909

281-5

7 08 1°47'00"

55° 51 '50'

11-99958

9-91788

2-08170

120-7

5 18 1°19'30"

55° 24' 20'

11-74157

9-91550

1-82607

67-0

2 27 36' 45"

54° 41' 35'

11-07136

9-91173

1-15963

14-4

21 5' 15"

54° 10' 05'

9-38117

9-90888

f-47229

0-3

3 09 47' 15"

53° 17' 35'

11-28965

9-90401

1-38564

24-3

5 24 1°21'00"

52° 43' 50'

11-75780

9-90080

1-85700 71-9

12 47 3° 11' 45"

50° 53' 05'

12-50621

9-88979

2-61642

413-4

15 31 3° 52' 45"

50° 12' 05'

12-67446

9-88553

2-78893

615-1

10) 1992-8

Mean value of w.

. 199

log cos p = 9-90031
log sin A0 = 9-88380
log 199 = 2-29885

log 121 = 2-08296
.-. Bm = 121" = 2' 01"

Mean value of angle between R.M. and star at elongation
= 83° 14' 21" - 2' 01" = 83° 12' 20"

140 ASTRONOMY FOR SURVEYORS.

The difference between the results of the two calcula-
tions is so small that clearly the more simple approximate
method is quite sufficient for the surveyor. If the com-
putation be made for the last four observations only, the
difference between the results of the two methods amounts
to 8" ', and for the last observation alone the difference
is 19". For the surveyor it is only necessary to use the
more accurate method of calculation for unbalanced
observations at a considerable time from elongation.

It may be proved that, provided the observations extend
evenly over an equal time on each side of elongation, there
is no need for the surveyor to know the local time with
great precision, an error of 1 minute in the time producing
an error of only about 1" in the azimuth.

But if the observations do not extend on each side of
elongation the case is different, and a more accurate
knowledge of the time is essential.

Second Method — Horizontal Angle and Altitude being Noted
at each Observation.— With the same notation as before, the
star being in any position, we have

cos p= cos c cos z-\- sin c sin z cos A.
Writing x = z — z0, this becomes

cos p = cos c cos (z0 -f x) + sin c sin (z0 + x) cos A .
p, c, and z0 being constants, this equation gives A as
an implicit function of x.

Differentiating the equation three times in succession,
the work being rather long but quite straightforward > we
find that when x— o

d A
- — =o,
dx

d2 A cot p
d x2 sin z0
d3 A 3 cot p cos z0

THE DETERMINATION OF TRUE MERIDIAN. HI
Therefore, by Taylor's Theorem

•.in z0 2

COt pOOB ».(£-..)» gin.r;

sin2z0

provided that A0 — A and z— z0 are expressed in seconds
of arc.

To get some idea of the relative values of the terms in
this series, we find, if the star observed has a polar distance
of 30° and the latitude is also 30°, then z0= 54° 44' 09",
and if z~ z0= 1°, the second term works out at 66" and
the last term to 0-8". If z — z0= 2° the values become
264" and 6" respectively.

The last term in (9) is equal to

coe'poosc o3,,

sin p (cos2 p — cos2 c)

and has, therefore, an infinite value if p= c, in which
case the star passes through the zenith. This is clearly
of no practical importance.

The following are the values of the last terms in different
latitudes for a star 30° distant from the celestial pole,
if z-s0 = 1°:-

Latitude. Value of Last Term in (9).

50°, 3-5"

40°, 1-5"

30°, 0-8"

20°, 0-4"

10°, 0-2"

0°, 0"

If z — z0 = 2° the preceding values should be multiplied
by 8.

It follows, therefore, that for the ordinary work of the
surveyor the correction involved in the last term of the
series is quite negligible for observations extending over

142 ASTRONOMY FOR SURVEYORS.

a range of altitude of 2°, or 1° on each side of elongation,
provided that the star does not pass within 10° of the
zenith. At places near the equator the observations may
clearly extend over a very much greater range of altitude
with the same degree of precision.

To determine over what range of time the observations
may extend, we find on differentiating the equation
cos z = cos c cos p + sin c sin p cos h

d z sin c sin p sin h

that - -= sin p for a star at elongation.

d h sin z

This = -I, if p=30°.

Thus, the rate of change of altitude at elongation does
not depend on the latitude, but simply on the polar
distance of the star, and for a star distant 30° from the
pole we have

dh= 2dz.

Therefore, if dz= 1°, dh= 120' of arc, or 8 minutes
of time, the altitude of the star near elongation thus
changes by 1° in about 8 minutes. For stars closer to
the pole the time taken for the same change of altitude
will be greater.

Practical Computation. — We conclude that for a set of
observations extending over a range of altitude of about
2°, or 1° on each side of elongation, occupying, in the case
of a star with a polar distance of 30°, about 16 minutes
of time, it is amply sufficient to use the formula

_^=co^(,-^sinr/
sin z0 2

It should be noticed that the error made by the use
of this formula in the final reduction of a set of obser-
vations will be very much less than the error made in the
reduction of the single observation furthest from elonga-
tion. We have based the stated limitations upon the
error made in the reduction of the single observation,

THE DETERMINATION OF TRUE MERIDIAN. 143

so that for a complete set of observations the time occupied
may be extended somewhat beyond the limits given above.
In low latitudes the observations may extend over a
greater range than in high latitudes. In latitude 10°,
for instance, the observations may extend over half an
hour, and formula (10) will still give the average result
of the set of readings correct within less than 1".

If the range of altitude is too great, or it is desirable
to compute A0 — A with the greatest precision possible,
then this value must be reduced if z>20, or increased if
2<z0, by the amount

cot p cos z0 (z— z0V

sin2 z0 2

sin2!". . . (11)

The computation by means of (10) is somewhat facili-
tated by making use of the same tables for circum-meridian
calculations as have been shown to be suitable for the
reduction by the first method. For since z — -z0 is a small
angle, we have, within the degree of accuracy to which
the tables are computed,

(? y V2 o «*in2 1 (? 9 \

\6 £Q) .. « aill 0 \6 ^Q)

-T~ sm ' ~^Tr-

• / \9

and consequently we can take the value of sin I"

straight from the tables.
Then, writing

cot p (z— z0)2 .

B= ,m= sin 1 ,

sin 20

we get for each observation, just as in the previous
method,

A0=A+Bm;

or. angle between R.M. and star at elongation

= observed angle between R.M. and star ± B m.

H4 ASTRONOMY FOR SURVEYORS.

Since B is a constant, we therefore get, on averaging
the whole set of observations : —

Mean angle between R.M. and star at elongation
= mean observed angle between R.M. and star
i B x mean value of m.

Whether the + or — sign is to be used depends upon
the position of the R.M. and upon which angle between
the star and R.M. is measured. It will be obvious in
any particular case which sign should be taken.

If the tables for m are not available, then it is better
to write

cot p sin 1"

B= ,m=(z-z0)2

sin z0 2

and proceed as before, this time computing m for each
observation. The use of the tables does not thus really
make very much difference.

A defective knowledge of refraction does not seriously
affect the accuracy of the work. For even if the altitude
is in error by 15", the resulting error in azimuth is only
about three-quarters of a second of arc.

The following example illustrates the method of reduc-
tion. It will be seen that the calculations are simple,
and the method is undoubtedly capable of much greater
accuracy than the ordinary methods of making elongation
observations :—

Star observed — a1 Crucis.

Eight Ascension — 12 hrs. 21 mm. 54 sec.

Declination— 62° 37' 47" S.

Date — March 5th, 1915.

Place — Burnside .

Latitude— 34° 55' 38" S.

Longitude — 9 hrs. 14 min. 36 sec. E.

Standard Meridian — 9 hrs. 30 min. E.

THE DETERMINATION OF TRUE MERIDIAN. 145

COMPUTED VALUES.

Standard time at elongation — 9 hrs. 13 min. 18 sec. p.m.
A0 = 34° 06' 25"
20 = 49° 51' 22"

Face.

Object.

Mean Vernier
Readings on
Horizontal
Circle.

Observed
Zenith
Distance.

= Observed
Zenith
Distance
Corrected for

z - z0.

HI.

Refraction.

R.M.

360°

Star

76° 56' 30"

50° 53' 00"

50° 54' 10"

62' 48"

34-36"

Star

76° 55' 30"

50° 26' 00"

50° 27' 09'

35' 47" !

11-15"

Star

76° 55' 00"

49° 57' 15"

49° 58' 23'

7' 01" !

0-43"

Star

76° 55' 15" !

49° 36' 00"

49° 37' 07'

14' 15" :

1-77"

Star

76° 55' 45"

49° 13' 45"

49° 14' 51'

36' 31"

11-63"

Star

76° 57' 00"

48° 38' 30"

48° 39' 34'

71' 48" i

44-92"

R.M.

360°

Mean value of m,

6 ) 104-26
. 17-38"

Mean observed angle between star and R.M. = 76° 55' 50"

B =

cot

tan 62° 37' 47"

= 2-527.

sin 20 sin 49° 51' 22"
Therefore, mean value of angle between R.M. and star at elongation

= 76° 55' 50" - 2-527 X 17-38"
= 76° 55' 50" - 44"
= 76° 55' 06"

EXAMPLES.

1. At a place in latitude 30° N., prove that the azimuth of a circum-
polar star having a declination of 80° N. when at Eastern elongation is
11° 34' 00-8", and that the hour angle of the star is then 84° 9' 25-3".

Find the time taken for the azimuth to decrease by 5".

Ans. 3 min. 34 sec.

2. At a place in latitude 30° S., prove that the azimuth of a circum-
polar star having a declination of 60° S. when at Western elongation is
215° 15' 51-8", and that the hour angle of the star is then 70° 31' 43-6".

Find the time that elapses before the azimuth is diminished by 5".

Ans. 2 min. 11 sec.

146 ASTRONOMY FOR SURVEYORS.

3. In latitude 37° S., the sun's declination being 14° S., show that at
9 a.m. the sun's azimuth is 72° 14' 39".

4. Compute the azimuth of a star having a declination of 75° S. when
at Eastern elongation, at a place in latitude 30° S.

Ana. 162° 36' 39-4".

5. Demonstrate that if two circumpolar stars A and B are in the same
vertical at some instant on the East of the meridian, A being above B,
they will later be simultaneously on the vertical making the same angle
on the West of the meridian, B being then above A.

6. At Greenwich noon, June 1st, 1914, the declination of the sun is 21°
58' 52-9" N., the variation in one hour being 20-96". At noon on June 2nd
the declination is 22° 07' 04-4", the variation in one hour being 20-00".
Find the sun's declination when the local time at a place in longitude 50° W.
is June 1st, 1914, 4 p.m.

An*. 22° 01' 25-5".

7. The corrected observed zenith distance of the sun on the afternoon of
March 17th at a place in latitude 34° 56' S. is 62° 19'. If the sun's declination
is 1° 28' S., compute its azimuth, to the nearest minute, of arc, at the time
of observation.

Ana. 289° 20'.

8. At a place in latitude 41° 12' 40" S. and longitude 11 hrs. 39 min.
34 sec. E. on the evening of the 15th January, 1913 (with the object of
checking a traverse bearing), the altitude and bearing of a second magnitude
star were observed through a break in the clouds. It was necessary to
compute the approximate R.A. and dec. of the star to identify it in the
catalogue, in order to obtain the precise elements for the calculation. From
the following data, find the star's R.A. and dec. : —

Star's true altitude, . . 43° 52' 34"

Bearing corrected for convergence, 131° 3' 14"

Sidereal time G.M.N., 15th Jan-
uary, ..... 19 hr|. 37 min. 19 sec.

N.Z, standard mean time (11 hrs.

30 min. E.), ... 8 hrs. 20 min. 51 sec.

Ans. R.A. = 8 hrs. 18 min. 54 sec.
Dec. - 54° 22' 11".

9. Determine the difference of azimuth of the sun at its rising in mid-
winter and mid-summer, also the difference (expressed in mean solar time)
in the lengths of the days at these two times. Assume the latitude of the

THE DETERMINATION OF TRUE MERIDIAN. 147

place to be 30° N., and the greatest declination of the sun 23° 27'. Disregard
corrections for refraction and parallax.

Ans. Difference of azimuth

= 54° 42'.

Difference of lengths of days
= 3 hrs. 52 min.

10. In latitude 30° 18' S., longitude 123° 40' E., the following sun obser-
vation was taken at 4 hrs. 45 min. p.m. : —

Alt., . . 22° 28' 30", 258° 43' 30" |O_ R.M.
Co-alt... . . 67° 55' 30", 258° 51' 30" "o I 357° 46'.

The sun's declination for the day, G.M.N., was 20° 19' 02" S., and for
the preceding day 20° 06' 16" S., the semi-diameter being 16' 14". Find
the true bearing of the R.M.

Ans. 328° 08' 08".

11. Find the bearing and altitude of a star at its Eastern elongation, also
the mean time of elongation. The latitude of the place is 31° S., the longitude
8 hours West, the R.A. of star is 6 hrs. 21 min. 30 sec., its declination 52°
37' S., and sidereal time at G.M.N. on the day of observation 14 hrs. 28 min.

Ans. Bearing, 134° 54'
Altitude, 40° 25'.
Mean time, 11 hrs. 38 min.
55 sec.

12. In latitude 25° 58' N. Polaris was observed at its Eastern elongation,
its declination for the date being 88° 44' 20". Compute the azimuth of the
star.

Ans. 1° 24' 10".

13. At Adelaide (latitude 34° 55' 38" S., longitude 9 hrs. 14 min. 20 sec. E.)
a forenoon observation was made of the sun on June 24th, 1914.

From two observations taken with F.R. the mean angle between R.M.
and sun was 85° 34' 05", the mean altitude 24° 03' 50". The mean time was
10 hrs. 7 min. 30 sec. a.m. (standard time of meridian 9 hrs. 30 min. E.).
With F.L. the mean angle between R.M. and sun was 87° 21' 00", the mean
altitude 24° 55' 07", the mean standard time 10 hrs. 15 min. 30 sec. a.m.
The sun's declination at G.M.N. on June 23rd was 23° 26' 51-9" N.. the
variation in one hour being 1 -26" on the 23rd, and 2-29" at noon on the 24th.
The angle between the sun and R.M. was measured from the sun to the right.
Determine th*1 true bearing of the R.M. Allow for refraction and parallax.

Ans. 118° 09' 28".

148

ASTRONOMY FOR SURVEYORS.

14. During the evening of the date 28th July, 1914, several bearings of
a Centauri were observed when it was near elongation. Find the true
bearing of the referring lamp, which was assumed to be 179° 00' 00".

OBSERVATIONS.

Statute Time,
10 Hours East of Greenwich.

10 hrs. 32 min. 15 sec.
10 hrs. 37 min. 20 sec.
10 hrs. 42 min. 06 sec.
10 hrs. 47 min. 12 sec.
10 hrs. 52 min. 05 sec.

Longitude 9 hrs. 39 min. 54 sec. E., Latitude 37° 49' 53" S.
R.A. of star 14 hrs. 33 min. 48 sec., Declination 60° 29' 16" S.
Sidereal time, G.M.N., July 28th, 1914. 8 hrs. 21 min. 13-93 sec.

Bearing of Weight of
ec Centauri. Observation.

217° 29' 10"

217° 32' 44"

217° 34' 48"

217° 36' 02"

217° 35' 25"

149

CHAPTER IX.

THE DETERMINATION OF LATITUDE.

THERE are many possible ways by which the surveyor
may determine the latitude of the place of observation,
but, as in the previous chapter, we shall here confine our
attention to the most practicable and most generally
used methods.

First Method — By Meridian Altitudes of Sun or Star. —
This is a very convenient and simple way of finding
latitude, where the greatest possible precision is not
required, and depends upon the fact we have already
discussed in Chapter III. that the altitude of the celestial
pole is equal to the latitude of the place of observation.
It follows that the latitude may be at once obtained by
observing the meridian altitude of a body whose declina-
tion or polar distance is known. This is the method
commonly used by the sailor at sea, the altitude of the
sun at apparent noon being observed with a sextant.
In Fig. 37, if O denote the position of the observer, Z the
zenith point, P the celestial pole, then if an object be
observed at S]5 we have AP=AS1— PS15 or latitude
— meridian altitude — polar distance. This might re-
present the position of a circumpolar star at its upper
culmination. If it were observed at lower culmination it
would be in the position S2, and in that case AP = A S2
4- P S2, or latitude = meridian altitude + polar distance.

In other cases the object observed may be on the
opposite side of the zenith to P. If E denotes the point
where the celestial equator intersects the meridian, the
body may be at S3 or S4. Since BE+PA=90°, it

150

ASTRONOMY FOR SURVEYORS.

follows that B E = the co-latitude. Then at S3 we have
B E = B S3 — E S3, or co-latitude = meridian altitude —
declination. When the body is in the position S4 its
declination will be South if the observation is made in
northern latitudes or north if the place is in South latitude.
In that case we have B E= B S4+ E S4, or co-latitude
= meridian altitude + declination.

Thus in all cases the latitude can be very simply obtained
provided that we know the declination of the celestial body.

The observed altitude must be corrected for refraction
as discussed in Chapter VII., and as the amounft of this
correction depends upon the pressure and temperature
of the air, it is necessary, if the correction is to be made
as accurately as possible, that thermometer and baro-

meter readings should be taken at the time of observa-
tion. Usually it will be sufficient to take the refraction
correction straight from the table of mean refractions,
without troubling to allow for the difference between the
actual temperature and pressure from that for which
the table of mean refractions is made out, because the
maximum change in the refraction due to an alteration of
temperature only amounts to about 3" per 10° F., and for
a change of pressure to about 5" per inch of barometer.

In the case of the sun, since it is the altitude of the
upper or lower limbs that must be observed, and it is
the altitude of the sun's centre that ia required, a correc-
tion must be made for its semi-diameter. Another

THE DETERMINATION OF LATITUDE. 151

correction also must be made to allow for parallax.
Both of these are found from the Nautical Almanac.
With observations upon the fixed stars neither of these
corrections is needed.

If the altitude of the sun is observed with a sextant
on land an artificial horizon must be used, in which case
the double altitude is measured. The following is an illus-
tration of such an observation, made in South latitude : —

Double altitude sun's lower limb,. . 64° 13' 10"
Index error + ... 4' 5"

2 ) 64° IT 15"

32° 08' 37-5"
Refraction — . 1' 55"

32° 06' 42-5"
Parallax + . 7"

323 06' 49-5"
Semi-diameter -f- . . 15' 50"

Altitude of sun's centre, . . 32° 22' 39-5"

Declination N, . . 19° 47' 53"

Co-latitude, . . 52° 10' 32-5"

.-. latitude = .... 37° 49' 27-5"

With observations upon the sun, if the local mean time
is known, the time of apparent noon may be found by
applying the equation of time as found from the Nautical
Almanac. The altitude of the sun's lower or upper limb
may then be observed at the proper instant as measured
by the watch. The effect of an error in time will depend
upon the latitude of the observer and the declination of
the sun. In latitude 45°, with the sun on the celestial
equator, an error of 1 minute in time will produce an
error of only 2 seconds in the measured altitude. Under
the same conditions if the time is wrong by as much as
10 minutes, the altitude measured will be too small
by 49 seconds. So that for the ordinary purposes of the
surveyor, when the observation is made in this way, it

152 ASTRONOMY FOR SURVEYORS

is not necessary to know the local time with great exact-
ness. If the approximately correct time is not known
the sun is followed by the observer^ and the altitude
measured when it attains its greatest value.

With observations upon the stars the same general
principles will apply. With close circumpolar stars it is
possible to take two observations for altitude upon the
same star, the face of the instrument being reversed after
the first reading is taken. When this can be done the
accuracy of the determination is increased, but as a rule
the altitude changes too rapidly for this to be possible.
In latitude 30°, for instance, the altitude of a star having
a polar distance of 30° is 48" less 5 minutes before and
after its culmination than when on the meridian.

Zenith Pair Observation of Stars. — A great improvement
upon the accuracy of simple meridian observations may
be effected by making observations upon two stars
which culminate at approximately equal altitudes on
opposite sides of the observer's zenith. The altitude of
one star having been observed at culmination, the face
of the instrument is reversed and the meridian altitude
of the second starts then measured. The two stars must,
of course, be chosen so that the second culminates at a
convenient interval after the first. The method is com-
monly referred to as that of latitude determination by
" zenith pair observations/' No attempt is made to
take two observations on the one star, and the combina-
tion of the two results largely eliminates errors of re-
fraction and errors due to the graduation of the vertical
arc. Thus, in Fig. 37, if Sj and S3 denote the two observed
stars, we obtain from the observation upon Sx

lat.= AS1-P-S1=AS1-plJ . . (1)
and from the observations upon S3

co-lat. = B S3 - E S3 = B S3 - (90° - p3)

lat.= 180°- B S3~ p* (2)

THE 'DETERMINATION OF LATITUDE. 153

Taking the average of the determinations (1) and (2), we
obtain

Thus, in the final determination it is the difference
of the measured altitudes A S, and B S3 that is required,
and as any error in the allowance made for refraction will
affect both the altitudes alike, the error will practically
disappear when we subtract them. Consequently, the
method almost eliminates errors due to an uncertain
knowledge of the refraction, and also enables instru-
mental errors to be largely eliminated by taking two
separate observations with opposite faces of the
instrument.

If the local time and consequently the sidereal time is
known with fair accuracy, the best way is to intersect
each star at the instant when the sideral time is equal
to the star's right ascension. This is found from the
Nautical Almanac, and the two stars will be selected for
convenience, if possible, so that their right ascensions
differ by from 10 to 30 minutes. If the time is not known
accurately, then the telescope must be directed to the true
meridian, and the altitude measured when the star
intersects the vertical wire. Readings must be taken
also of the barometer, thermometer, and alidade
level.

The following example is taken from the Western
Australian Handbook for Surveyors : —

Date, 1st May, 1910.

6 Argus — observed altitude (South) = 58° 01' 30".
Alidade level - 0 = 5-8, E = 3-2.
1 Division of level = 15".

Barometer — 30-52". External thermometer = 72-5°. Attached ther-
mometer = 71°.
Note. — O means object end of telescope. E means eye end of telescope.

154

ASTRONOMY FOR SURVEYORS.

Compute refraction from " Bessel's Refractions," as given on pp. 430, 431
Chambers' Log Book, from the formula : —

True ref . = mean ref. x B x t x T.

Mean refraction alt. 58° 01' = 36-1", . . 1-55751

B for 30-52" = 1-032", . 0-01368

<for71°F. = 0-997", . 9-99870

T for 72-5° F. = 0-955", . 9-98000

True refraction ^35-5" 1-54989

Obsd. alt.
Ref.

Level +

True alt.
Polar distance

= 58° Or 30"
35-5'

58° 00' 54-5'
19-5'

= 58° 01' 14"
= 26° 04' 19-3'

Level.
O 5-8
E 3-2
2 )J_-6

1-3 x 15" - 19-5'

Latitude = 31° 56' 54-7"

Date, 1st May, 1910.

I Leonis — Obsd. Z.D. (North) = 42° 57' 00".
Alidade Level, 0 = 3-5, E = 5-5.
Barometer = 30-55", External Thermometer =•- 72-0°.

Attached Thermometer = 71-0°.

Logs.

Mean ref. alt. 47° 3' = 53-7", . . 1-72997

B for 30-55" 1-032, . . . 0-01368

* for 72-0° = 0-997," . . . 9-99870

T for 71-0° = 0-956", , . 9-98046

True refraction == 52-8",

Obsd. alt.

= 47° 03' 00"

Ref.

52-8"

47° 02' 07-2"

Level

15-0"

. 1-72281

Level.
E 5-5
0 3-5
2 )_2-0

TO X 15" = 15

True alt. = 47° 01' 52-2"

Declination =11° 01' 16-1"

Co-latitude = 58° 03' 08-3"
Latitude = 31° 56' 51 -7"

Deduced latitude — 6 Argus (South),
I Leonis (North),

Mean

31° 56' 54-7"
31° 56' 51 -7"

31° 56' 53-2"

THE DETERMINATION OF LATITUDE. 155

Meridian Altitudes of a Star at both Lower and Upper
Culminations. — If the meridian altitudes of a star be
observed at both lower and upper culminations, then,
if these be separately corrected for refraction, the mean
of the two altitudes will give the altitude of the celestial
pole, which is equal to the latitude of the place. The
method does not require a knowledge of the decimation
of the star, but as this information is always to be obtained
in the Nautical Almanac, there is no practical advantage
to the surveyor, save perhaps in very exceptional cases.
On the other hand, the long interval necessary between
the two observations is a very practical inconvenience.
Consequently, the method is not one in practical use
amongst surveyors, although it is employed by astronomers
at fixed observations.

Second Method — By Circum-Meridian Observations. — Obser-
vations of stars or the sun taken near to the meridian
are commonly spoken of as circum-meridian observations.
By taking a series of altitudes of a star or the sun for
some few minutes both before and after it crosses the
meridian, instrumental errors may be largely eliminated,
and by proper methods of reduction the results may
be used to give a very accurate determination of latitude.
It is necessary to have the means of accurately noting
the time of each observation, and then each altitude
may be corrected or reduced so as to give us the corre-
sponding altitude on the meridian itself. Thus a series
of " circum-meridian " altitudes becomes equivalent to
a series of measurements taken on the meridian itself,
and in the taking of such a set of observations the instru-
ment may be reversed and its errors eliminated in a
way that is not possible with a single meridian obser-
vation. Still greater precision may be attained by
taking such observations upon equal numbers of stars
North and South of the Zenith, at approximately equal
altitudes.

156 ASTRONOMY FOR SURVEYORS.

In Fig. 38, let Z be the Zenith, P the celestial pole,
and S the observed star. As this is to be near the meridian,
the angle S P Z will be small.
Let z= SZ, the Zenith distance,

p = S P, the polar distance of the star,
c = P Z, the co-latitude,
t == the hour angle S P Z.
Then, from the triangle S P Z,

cos 2= cos c cos p-\- sin c sin p cos t. . (1)
Let x be the correction that has to be applied to the
observed zenith distance, z, in order to deduce the zenith
distance when the star is on the meridian.

Fig. 38.

Then meridian zenith distance = z— x= p — c.

t
If, now, in equation (1) we write cos t= 1—2 sin2 -,

we get j

cos z = cos (c — p) — 2 sin c sin p sin2 -.

cos z— cos (2 — x) = — 2 sin c sin p sin2 -.

x z\ t

2 sin - sin ( 2 — - j = 2 sin c sin p sin2 -.

sin - =

sin c sin p sin2 -

2 . / x

sin ( - -

THE DETERMINATION OF LATITUDE. 157

If x is small, we may now replace sin - by the circular

measure of \x, which is \x sin I", provided that x is
measured in seconds of arc. Also, we shall make very

/ X\

little difference to the result if, instead of the sin (z — -j

of the denominator we write sin (z— ;r)=sin (p— c).
Thus, if x is the correction to be applied in seconds of
arc, we obtain

2 sin2
sin c sin p 2

sin (p— c) sin I"

or, in the form in which it is more usually written, if a
denotes the observed altitude, A the altitude on the
meridian, / the latitude, and n the declination,

2 sin2 -
cos / cos n 2

A _ xy _ _ . , __

cos A sin I"

It will be noticed that if a series of observations are
taken upon the same star, the first factor in this expres-

cos I cos n

sion, i.e. - - — , is the same for them all. We will
cos A £

2 sin'-

denote this by B. If we write m = - , we have

sin 1"

A= a + B m.

The value of m in seconds may be computed, knowing
the value of t, or more conveniently it may be taken from
tables such as are given in Chauvenet's Astronomy or
from the abbreviated table given at the end of this
chapter.

Thus, if ttj, a2, a3, etc., denote a series of observed
circum-meridian altitudes of the same star, and ml5 ra2,

158 ASTRONOMY FOR SURVEYORS,

m3, etc., are the corresponding values of m, we obtain
a series of values for the corresponding meridian altitudes
given by the equations

AI= »i+ B ml
A2= «2+ B ra2
A3 = a3+ B m3, etc.

Therefore, if we denote by A0 the mean of the deduced
meridian altitudes, by a0 the mean of the actual observed
altitudes, and by m0 the mean of the computed factors
m, we have

A0 = a0 + B w0.

With the aid of tables for m, the reduction of the
observations thus becomes extremely simple. We take
the mean of the values of m, multiply by B, and add the
product to the mean of the observed altitudes.

The deduced mean meridian altitude is then corrected
for refraction and the latitude is computed as an ordinary
meridian altitude observation.

The value of B involves both the latitude and the
meridian altitude, since

cos I cos n
cos A.

but the value of I used in this is the approximate latitude
as deduced either from the map or from a simple meridian
observation. The value of A used is the meridian altitude
computed from the approximate latitude and the known
declination of the star. The approximate value of B
thus deduced is quite sufficiently accurate, when multiplied
by m, to give the correction required. A still higher
degree of accuracy may be attained by repeating the
calculation, using for B the value of the latitude as first
computed.

Before starting the actual observations, it is necessary
to calculate the time of the star's meridian transit. The

THE DETERMINATION OF LATITUDE. 159

observations should then be made within about ten minutes
on each side of this. The t in the formula is the interval
of sidereal time between the instant of actual observation
and the instant of meridian transit, expressed in angular
measure at the rate of 15° per hour.

The method involves an accurate knowledge of the
local time, and is then capable of a high degree of pre-
cision. To get the best results the errors should be
balanced by taking an equal number of observations on
stars both North and South of the Zenith. An equal
number should be selected on each side at approximately
equal altitudes. The errors are likely to be greatest for
stars observed near to the Zenith, especially when the
place of observation is near to the equator. The range
of observed altitudes should, if possible, lie between
40° and 75° above the horizon, and the closer the stars
are observed to the meridian the better will be the
results.

More Exact Methods of Reduction of Circum - Meridian
Observations. — The approximate formula that we have
given is the one usually adopted for the reduction of
circum-meridian observations. A still closer approxi-
mation may be obtained by using the more elaborate
formula

A=a+Bra+Cra',

2 sin4 \ t

where C= B2 tan A and m' '= '—,

sin 1"

A and B having the same significance as before.

The correction introduced by the third term in the
formula is usually very small when the observations are
made close to the meridian. If the value of t in minutes
does not exceed two-fifths of the Zenith distance of the
star in degrees, then it can be shown that the correction
introduced by the term C mf is never more than V,

160 ASTRONOMY FOR SURVEYORS.

so that the more exact formula is only required where
the highest precision possible is sought.

This may be obtained in a manner similar to that
employed in Chap. VIII. for the corresponding formula
for circum-elongation observations for azimuth.

The Limits of Time for the Observations. — According to
what we have just seen, the greatest interval of time in
minutes between any observation and the instant of
meridian transit should not exceed two-fifths of the
zenith distance of the star in degrees if the error in re-
ducing the observation to the meridian is to be limited
to 1". It is not possible to work so precisely as this
with the instruments commonly used, and the time may
be extended somewhat beyond this limit. In general,
it seems a good rule to say that the greatest value of
t in minutes of time should not exceed one-half of the
zenith distance in degrees. Thus, if the altitude of the
star is 50°, the observations may be made within
20 minutes on each side of the meridian transit. In
that particular case the maximum error would still only
amount to 1", but in other cases the error may be
somewhat greater if this rule is followed, but never so
much as 3", provided that the star is not within 10° of
the zenith.

Circum-Meridian Observations of the Sun. — As a general
rule, it is more convenient for the surveyor to make
observations upon the sun than upon the stars, and
exactly the same method as we have described may be
followed for circum-meridian observations of the sun.
Obviously the sources of error cannot be balanced in the
same way as with stars by taking observations both
North and South of the. zenith, so that such precise work
is not possible. There is another difficulty arising from
the fact that the sun's declination is not constant and,
if the observations extend over 30 minutes, it may vary
by as much as 30". If, however, a similar number of

THE DETERMINATION OF LATITUDE.

161

observations are made both before and after apparent
noon, the errors will very nearly balance in the mean,
provided that in the computations the value of the
declination used is the value at apparent noon. This is
not exact, but sufficiently so for all but the most precise
work.

An even number of observations should be made,
usually eight.

The first observation will be to the sun's upper limb
with F.R. Then two in succession to the lower limb
with F.L., next two in succession to the upper limb, the
instrument being reversed, once more with F.R. Two
more to the lower limb with F.L., and finally one to the
upper limb with F.R. With this order the sun's diameter
is eliminated in the mean. The alidade level should be
read and recorded at each observation. The method of
recording and the calculation is shown in the accom-
panying example : —

EXAMPLE OF CIRCUM-MERIDIAN OBSERVATION OF SUN FOR LATITUDE.

Place, .

Longitude,

Date,

Survey Office, Adelaide.
9 hrs. 14 min. 20-3 sec.
July 4th, 1914.

Sun's

Face of

Standard

Vertical Circle.

Observed.

ment.

Time.

Mean.

H. M. S.

12 12 58

32° 21' 45'

32° 21' 30'

32° 21' 37"

12 14 53

31° 51' 30'

31° 51' 30"

12 16 57

31° 52' 00'

31° 52' 00"

12 19 00

32° 23' 00'

32° 23' 00"

12 20 00

32° 23' 00'

32° 23' 00"

12 21 56

31° 52' 00'

31° 52' 00"

12 23 58

31° 51' 50"

31° 51' 50'

31° 51' 50"

12 25 56

32° 21 '50"

32° 21 '50'

32° 21' 50"

Mean observed altitude = 32° 07' 05-9".

162

ASTRONOMY FOR SURVEYORS.

rf '
1

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OS OS O OS 00

OS 05 0 OS 6

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I a 1 M

ta § 2

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02 02 o

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For Time of Apparent Noon.

.? ? 88^ os

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g Th O CO CO »O OS

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•g - »•'•<- I H

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eg o> o C~i

& S 1 s. S 1

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For Approximate Latitude.

vv vTh CO i^H t^-CO

OCO t^-O i-HC*5 4<IC

O<M COTH O<M F-HTt<

CC^H »M «3 IOOO ^*1O
«<l <M I-H O IO O »O

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(M (M <M <N U3 ^

CO C*5 CO 'M »O CC

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' • • § 1 •

v 2 ^ G 0

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• a T3 C o £ ,2 3
T3 3 e £ g 3 a .13

1" « T 5 1 a 3

THE DETERMINATION OF LATITUDE.

COMPUTATION FOR LATITUDE.

163

Mins. Sees,
6 40
4 45
2 41
38
22
2 18
4 20
6 18
8"

87-3'
44-3'
14-1'
0-8'
0-3'
10-4'
36-9'
77-9'

) 272-0'

m0t .... 34 -0"
m0 B, . . . . 30-3"
Mean observed altitude, . 32° 07' 05-9"

An,
Refraction —

Parallax + .

Corrected altitude,
Declination, .

Co-latitude, .
Latitude.

. 32° 07' 36-2"
1'31"

32° 06' 05-2"
01"

. 32° 06' 12-2"

. 22° 58' 23"

. 55° 04' 35-2"
. 34° 55' 25"

Third Method — Latitude by Prime Vertical Transits.—

The Prime Vertical has been already defined as the
vertical plane at right angles to the meridian, running
truly East and West. Stars with polar distances less than
90° and greater than the distance of the pole from the
zenith — i.e., greater than the co-latitude of the place —
will cross the prime vertical twice in a sidereal day. If
the interval of time between the East and West transits
of a star be measured, and the decimation of the star be
known, then the latitude can be readily computed. Thus,
in Fig. 39, let E Z W represent the prime vertical of the
observer, Z being the zenith. Let P be the celestial pole
and A C B the portion of the star's path described on the
same side of the prime vertical as the pole. A and B are
the points where the star's path intersects the prime

164

ASTRONOMY FOR SURVEYORS.

vertical. If A and B respectively joined to the pole P
by arcs of great circles, P A and P B will each be equal
to the star's polar distance or to the complement of its
declination. Then, in the spherical triangle A Z P, the
angle at Z is a right angle, P Z = the co-latitude, P A
= the star's polar distance, and the angle A P Z, if turned
into time at the rate of 15° per hour, will represent half
the interval between the transits at A and B measured
in sidereal time.

From Napier's Rules we have

cos A P Z = tan P Z x cot A P,
whence tan latitude = tan declination x sec t,

where t = half the interval of sidereal time between the
transits expressed in angular measure.

By this method the errors due to uncertainty with regard
to refraction are largely eliminated, because the times of
transit are observed instead of altitudes. The method
does not require a knowledge of the exact local time, as
it is an interval of time that has to be measured, conse-
quently it is sufficient for the surveyor to have a watch or
clock whose rate is known.

It will be obvious that in places where the elevation
of the celestial pole is small — that is to say, in places

THE DETERMINATION OF LATITUDE. 165

near the equator — the paths of such stars as move across
the prime vertical will intersect it very obliquely, and it
will not be possible to secure a good determination of the
exact time of intersection. A precise measurement will
be more easy in places of higher latitude.

The Effect upon the Determination of an Error in the Measure-
ment of the Time Interval. — To make a determination of
latitude by the method just described, the surveyor has
to set out the direction of the prime vertical, and also to
measure the time interval between the East and West
transits. In order to judge therefore of the degree of
precision of which the method is capable we require
to investigate the effect of small errors in each of these
measurements.

If we denote the latitude by I and the declination of
the observed star by d, we have

tan /= tan d . sec t. (1)

If a small error y is made in the measurement of t,
and x is the corresponding error made in the latitude,
tan (/+ x) = tan d . sec (t+ y).

Expanding and writing tan x= x, cos y = 1 , sin y = y
' since x and y are small, we have

x + tan I tan d

1 — x tan I cos t — y sin t

.-. neglecting the product of the small quantities x y,
we get

x cos t -f- tan I cos t — y tan / sin t = tan d — x tan d tan I.
Making use of (1), this becomes

tan2d
=< tan

tan d sec2 I // tan2 d

sin 2 I /tan2 1

166 ASTRONOMY FOR SURVEYORS.

The student who understands differential calculus can
obtain this result at once by differentiating equation (1),
keeping d constant.

From this equation we get the important practical
deductions that if d is nearly =1, x will be very small,
and that if d is nearly = 0, x will be very large. So that
it would seem that the stars most suitable for observation
are those whose declinations are nearly equal to the
latitude. A star having a declination the same as the
latitude would pass through the zenith point, and the
declination must be somewhat less than the latitude for
the method to be possible. On the other hand, a star
with zero declination would pass through the E. and W.
points on the horizon at the prime vertical for all latitudes,
the interval of time between its transits would be exactly
six hours no matter what the position of the observer,
and no determination of latitude could be made. It
would apparently follow, then, that the best stars to select
are those that cross the prime vertical near the zenith.
But a star crossing the prime vertical very near to the
zenith intersects it so obliquely that it is not possible
to make an accurate determination of the time of transit.
The distance from the zenith, at which the path of the
star will make a sufficiently large angle with the prime
vertical to enable a good measurement of the transit
to be made, will depend upon the latitude of the observer.
And the practical conclusion is that the stars observed
should be as high up on the prime vertical as is consistent
with an exact determination of the time of transit. Stars
which cross it low down must be avoided, as they lie
near the celestial equator, and the error in latitude
produced by a slight error in time is then very large.

A definite calculation will give a better idea of the
effect of a defective measurement of the time interval.
If we take a place in latitude 30°, and suppose the obser-
vation to be made on a star with a declination of 10°,

THE DETERMINATION OF LATITUDE. 167

then x= l-3y. Now t in our formula is half the total
time interval between the transits, so if this whole interval
is in error to the extent of one second of time, y = half a
second. But half a second of time is equivalent to 7-5
seconds of arc, and this multiplied by 1-3 gives 9-7 seconds
of arc as the error in latitude caused by an error of one
second in the time interval.

If in the same latitude the star observed has a declina-
tion of 20°, then, from the same formula, x== -52 y. In
this case a mistake of 1 second in the total time interval
will cause an error of 3-9 seconds in the latitude. If the
decimation is 25°, x= -32 y, and the corresponding error
in latitude is 2-4 seconds. In higher latitudes the errors
are still greater.

Clearly, even if the surveyor is to be content with a
determination of latitude to the nearest minute of arc,
he must be able to rely upon his measurement of the time
interval within a few seconds.

The Effect of an Srror in the Direction of the Prime
Vertical. — The error arising from a defective setting out
of the prime vertical is not nearly so serious, because,
if this is marked out so that the time of the Eastern transit
of the star is earlier than it should be, then the time of
the Western transit will be correspondingly hastened,
so that the interval between the transits will be very
little different to that when the prime vertical is correctly
located. Thus, in latitude 30°, the measurements being
made on a star with a declination of 20°, even if the
prime vertical is set out as much as 1° out of its true
position, the resulting error in the latitude determination
is less than 1 minute of arc. So that a comparatively
rough determination of the prime vertical is sufficient
for the surveyor's purpose. It is, of course, most important
that the instrument shall be in accurate adjustment, so
that it will sweep out a truly vertical circle. But instru-
mental errors may be largely eliminated by taking

168 ASTRONOMY FOR SURVEYORS.

observations on alternate nights with the instrument
reversed.

Although the method is capable of giving results of
great precision, the practical inconvenience caused by
the long interval between transits and the necessity for
exact time measurements rather put it out of court as
a suitable method for ordinary surveyors in the field.

The same method may be applied, with some modi-
fication of formulae, to any vertical circle whatever. But
the prime vertical circle is the most suitable for accurate
work.

Striding Level Correction to Prime Vertical Observations. —
The striding level should always be used with prime

vertical observations as the resulting determination of
the latitude is in error by an amount equal to the angle
which the transverse axis of the telescope makes with
the horizontal. Thus, in Fig. 39a, if P denotes the celestial
pole, Z the zenith, and E W the East and West points on
the horizon, then, if the striding level shows an error in
the horizontality of the transverse axis of the telescope,
the circle upon which the observations are actually made
will be E C W instead of the true prime vertical E Z W.
The star is observed to transit at the point S, the angle
S P C = t, and the angle S C P is a right angle.

THE DETERMINATION OF LATITUDE. 169

Thus, we shall get

cot C P = tan declination x sec t.

The true co-latitude is then C P ± Z C, the + sign being
taken if, as in the figure, C is on the same side of Z as P,
and the — sign being used if C and P are on opposite
sides of Z. This is determined by the direction of the level
error, and Z C = the angular measure of the level error.

Thus, to make the correction, the computation for
latitude is made in the ordinary way, and then we add
or subtract the striding level error.

EXAMPLE. — At a place in S. latitude the interval between the passage of
Sirius across the prime vertical is 6 hrs. 09 min. 19-1/3 sec. mean time. The
mean readings of the bubble on striding level were 10 N. and 14:8., each division
being = 20". The declination of the star is 16° 35' 33" 8. Determine the
latitude.

6 hrs. 09 min. 19*1/3 sec. of mean time
= 6 hrs. 10 min. 20 sec. of sidereal time
= 92° 35' 00" of arc
tanlat. = tan dec. X sec. 46° 17' 30".

tan dec., 9-4741732

cos 46° 17' 30", . 9-8394702

9-6347030

.-. lat. = 23° 19' 37".
But the striding level error necessitates a correction

= ?±.-t..10x 20 = 40".

As the South end of the transverse axis is the higher, the derived latitude
is too small.

.-. corrected latitude = 23° 20' 17" S.

Fourth Method — By the Altitude of the Pole Star at any
Time. — Provided that the exact local time and the
approximate longitude are known, the latitude may be
found from an altitude observation of a close circum-
polar star at any time. In the Northern Hemisphere the
Pole Star is commonly selected for this purpose, and
special tables are given in the Nautical Almanac for
reducing the observations. In the Southern Hemisphere

170 ASTRONOMY FOR SURVEYORS.

unfortunately there is no bright star sufficiently near
to the Pole to make the method a convenient one for the
surveyor.

In Fig. 40, let S be the circumpolar star, Z the zenith,
and P the pole as before. Then, with the previous nota-
tion, if

z = S Z, the zenith distance,

p= S P, the polar distance of the star,

c= P Z, the co-latitude,

t = the hour angle S P Z.

From the triangle S P Z we have

cos z= cos c cos p+ sin c sin p cos t,
or, if a is the observed altitude, and I the latitude
sin a = sin I cos p-\- cos I sin p cos t.

Fig. 40.

Now a will differ from I by a small quantity, which is
always less than p. In the case of the Pole star p is also
small, being about 1° 10' at present. Let

a= l-\- x,

where x is a small correction.

. •. sin I cos x + cos I sin x = sin / cos p -f cos I sin p cos t.

.-. sin 1(1- -+ ...)+ cos I (x- —+...)
£ b

«2
= sin/l— — ... cos

THE DETERMINATION OF LATITUDE 171

Neglecting the square and higher powers df x and p
in this equation, we get x= p cos t, which is the value
of # to a first approximation.

Next, retaining the squares of x and p, but neglecting
the higher powers, we get

p2 ^.2

x cos 1= p cos / cos t sin I -\ — sin I.

Substituting for x2 the value p* cos2 1, we obtain then
as a second approximation

x = p cos t — J tan / sin2 1 . p2.

The second term in this expression is very small, and
as tan / differs from tan a by only a small quantity, the
difference when multiplied by p2 will be too small to take
into account, so that we may write

x = p cos t — | tan a sin2 1 p2.

In this formula x and p are in circular measure, but
if x and p are measured in seconds we may write

x = p cos t — 4 p2 tan a sin2 1 sin 1",
so that we have for the latitude

1= a— p cos t + \ p2 tan a sin2 1 sin \" .

The formula is, of course, an approximation only, but
it can be shown that it is sufficiently accurate to give the
result within 1" of the truth.

To determine t, the sidereal time must be known
accurately at the moment of observation, and t is
then the difference between the sidereal time and
the right ascension of the star turned into angular
measure .

Four altitudes should be taken in as quick succession
as possible, one with F.R., two with F.L., and then again
one with F.R., the alidade level being read at each obser-

172

ASTRONOMY FOR SURVEYORS.

vation, and the chronometer times noted. The mean
of the altitudes and the mean of the chronometer times
are then taken as the basis for the reduction as a single
observation.

A Rough Method for the Determination of Latitude by
Noting the Rate at which Altitude of Sun or Star Changes
near the Prime Vertical. — This is only a very rough and
approximate method at best, but it is interesting because
of its simplicity, and because it requires no knowledge
of either the local time or the declination of the body
observed. But it is not to be classed along with the
previous methods.

In Fig. 41, let Z be the zenith point, P the celestial

Fig. 41.

pole, and R S two consecutive positions of the sun or
star. The change of altitude will be measured by the
difference between the arcs Z R and Z S, and the interval
of time between the two positions will be measured in
angular measure by the angle S P R.

In the triangle Z P R, Z R = zenith distance = z,
P Z = co-latitude = c, P R = polar distance = p, R Z P
= azimuth measured from elevated pole = A, Z P R
= hour angle = B.

In the triangle Z P S, suppose that z has become
changed to z — y, and B to B — x, c and p remaining
unaltered.

THE DETERMINATION OF LATITUDE. 173

Then from the formulae of spherical triangles, we have
cos z = cos p . cos c + sin p . sin c . cos B, and cos (z — y)
=,cos p cos c-\- sin p sin c cos (B — x).

Subtracting these expressions, and regarding x and y
as small quantities, so that we may write cos x = I, sin x
= x, etc., we obtain

y sin z = sin p sin c sin B . x.
sin z sin p

.out — = .

s in B sin A

y = x . sin c sin A,

or cos . latitude = - cosec . azimuth .

Thus, in order to determine the latitude, all we have to
do is to measure the change of altitude y that takes
place in a given time whose angular measure is x. If
t=the interval of time in seconds, x= 15 t seconds of
arc.

As the ratio — is to be multiplied by cosec A, and

the observation is made near the prime vertical, an error
in the azimuth A will have but a small effect upon the
result.

A convenient way of making the observation is to
take the time required by the sun, in the afternoon or
early morning, to cross the horizontal wire of the telescope,
observing at the same time the sun's approximate bearing.
For an afternoon observation, bring the sun's lower limb
into contact with the wire and start the stop watch.
When the sun is about bisected by the wire, read the
approximate azimuth of its centre. Stop the watch at
the instant that the upper limb becomes tangent to the
wire.

174 ASTRONOMY FOR SURVEYORS.

EXAMPLE. — At a place in South Latitude on March 17th, the sun took
2 min. 46-4 sec. to transit the horizontal wire of a theodolite, the bearing
of its centre being 289° 20'.

Diameter of sun = 32' 11-3" = 1,931-3"
15 X 2' 46-4" = 15 x 166-4 = 2,496

1 Q^l -^
cos lat. = 2496 X °OSeC 109° 20/'

1,931-3, 3-2858497

cosec 109° 20' = sec 19° 20', . . 10-0252082

13-3110579
2,496, . . 3-3972446

cos 34° 55', ..... 9-9138133
Therefore, the latitude is 34° 55' S.

It will be found on trial in this example that if the
azimuth is 1° out, the computed latitude is about 30'
in error, and we must know the azimuth of the sun within
2' if we wish to find the latitude to the nearest minute.
If the observation had been made with the sun nearer
to the prime vertical, however, an error in azimuth
would not produce anything like so serious an
effect.

To get anything like accurate results, the time must
be measured with great precision. In the above example
an error of one whole second in the time causes an error
of nearly three-quarters of a degree in the latitude. With
a stop-watch the time may be estimated to the tenth of
a second, but it is evident that only approximate
determinations of latitude are possible by this
method with the instruments at the disposal of the
surveyor.

The method is of interest, because it may be practised
upon a star without the use of any Nautical Almanac
Tables. It will give best results in high latitudes with

THE DETEKMINATION OF LATITUDE. 175

observations made as near to the prime vertical as
possible.

There are many other methods by which latitude may
be determined, but for the most part they are not so
convenient nor do they allow of the same elimina-
tion of instrumental errors as the four standard methods
described. The following is an illustration of a
method in which horizontal angles only have to be
measured : —

Determination of latitude by the Measurement of the
Horizontal Angle between Two Circumpolar Stars at their
Greatest Elongations one on each Side of the Meridian.

Let c be the co-latitude, pl and pz the respective polar
distances of the two stars, Ax and A2 the azimuths at
elongation, one being measured to the East and the
other to the West.

The measured angle = Ax + A2.

Then sin pt = sin c sin A1? . . (1)

and sin p2 = sin c sin A2. . . (2)

(1)+ (2) gives

z Pi— P-2 AI + A2 Ax— A2

2 sm - - cos - - = 2 sin c sin - cos - - .

22 22

(1)- (2) gives

Pi + Pz • Pi— Pz A! + A2 . Ax — A2

2 cos - sin '-- = 2 sm c cos - - sin - — -.

22 22

Dividing one equation by the other gives

pi~r pz j. Pi Pz Aj-f" A2 Aj A2

tan — - cot - = tan - — cot — .

22 22

176

ASTRONOMY FOR SURVEYORS

Since AX+A2 is known, this enables A1— A2 to be
computed. Hence Aj is found.

Then

sin c =

sin
sin

Example, taken, from Handbook of Instructions to South Australian Sur-
veyors.

Observed horizontal angle 77° 45' between Canopus and /? Tri. Aus. at
opposite elongations, polar distances 37° 22' and 26° 57'.

Z2 -tan 5° 12'

tan -2 = tan 38° 52' 30",

tan

-tXt — •£*<>

tan — 1-2~- •*
A, -A,

tan 32° 09' 30",

= 6° 40'

8-9597747

9-9064310

18-8662057
9-7984562

9-0677495

and -L~^Z =38° 52' 30"

Aj = 45° 32' 30"

sin P! = sin 37° 22', .
sin Aj = sin 45° 32' 30",

cos lat.,
.-. latitude = 31° 45' 20".

9-7831268
9-8535522

9-9295746

TABLE GIVING VALUES OF m FOR REDUCTION OF CIRCUM-MERIDIAN
OBSERVATIONS.

2 sin* ~

The values of m are given in seconds of arc.

THE DETERMINATION OF LATITUDE.

177

Additional Seconds of Time.
Value of t ,

of Time.
0

0 0-0

0-1

0-2

0-5

0-9 1-4

I 2-0

2-7

3-5

4-4

5-4 6-6

2 7-8

9-2

10-7

12-3

14-0 15-8

3 17-7

19-7

21-8

24-0

26-4 28-8

4 31-4

34-1

36-9

39-8

42-8

45-9

5 49-1

52-4

55-8

59-4

63-0

66-8

6 70-7

74-7

78-8

83-0

87-3

91-7

7 96-2

100-8

105-6

110-4

115-4

120-5

8 125-7

130-9

136-3

141-8

147-5

153-2

9 159-0

165-0

171-0

177-2

183-5

189-8

10 196-3

202-9

209-6

216-4

223-4

230-4

11 237-5

244-8

252-2

259-6

267-2

274-9

12 282-7

290-6

298-6 306-7

315-0

323-3

13 i 331-7

340-3

349-0

357-7

366-6

375-6

14 384-7

393-9

403-3

412-7

422-2

431-9

15 441-6

451-5

461-5

471-5

481-7

492-0

16 502-5

513-0

523-6

534-3

545-2

556-1

17 567-2

578-4

589-6

601-0

612-5

624-1

18 635-9

647-7

659-6

671-6 683-8

696-0

For intermediate values of t the corresponding values of m may be found
by simple interpolation.

EXAMPLES.

1. At a place in latitude North, the true zenith distances of a Cephei
(declination 61° 58' 21-1") is determined as 26° 54' 28-3" N. The zenith
distance of a Aquike (declination 8° 29' 22-7") is found as 26° 34' 27-5" S.

Find the latitude of the place.

Ans. 35° 03' 51-5".

2. In latitude 30° S. the times of transit of a star whose declination is
20° S. are observed across the prime vertical. If the direction of the prime
vertical is in error by 1°, show that the measured interval of time will be
too great by about 14 seconds.

3. An observation made in Antarctica on November 19th, 1912, gave
the altitude of the sun's centre as 42° 07-8', the temperature being 17° F.
and the barometer reading 27-2 inches. Correct for refraction and parallax,
and compute the latitude of the place, given that the sun's declination is

19° 21-6' S.

Ans. 67° 14-7' S.

178 ASTRONOMY FOR SURVEYORS.

4. The declination of the sun being 20° 39-9' S., its meridian altitude is
observed as 43° 17'. The correction for refraction and parallax being
— 00-9', determine the latitude of the place.

Ans. 67° 23-8' S.

5. The sun is observed on the prime vertical, morning and afternoon,
the times by watch being 7 hrs. 30 min. and 4 hrs. 14 min. The sun's
declination is 17° 31' 30". Compute the latitude.

Ans. 37° 17' 30".

6. At a place in S. latitude the interval between the passages of Sirius
across the prime vertical is 6 hrs. 9 min. 19£ sec. mean time. The mean
readings of the bubble on striding level were 10 N. and 14 S., each division
being = 20". The declination of the star is 16° 35' 33" S. What was the
latitude of the place of observation ?

Ans. 23° 20' 17" S.

7. The hour angle of Aldebaran (dec. 16° 20' 15" S.) when on the prime
vertical was found to be 4 hrs. 35 min. 19-5 sec. What was the latitude
of the place of observation ?

Ans. 39° 04' 3" S.

8. At a place in the Southern Hemisphere yz Ceti (dec. 2° 51' 22" N.)
was observed at equal altitudes of 48° 02' 20", and the interval in mean
solar time between the two occurrences was 16 min. 12 sec. Required the
latitude of the place.

Ans. 43° 50'.

9. Antares crossed the prime vertical at 13 hrs. 52 min. sidereal time.
Find the latitude of the place of observation.

R.A. of Antares, 16 hrs. 23 min.

Dec. „ 26° 13' S.

Ans. 31° 54' 49" S.

10. The altitudes of a star when it crosses the meridian and prime vertical
are respectively 65° and 10° (corrected). Find the star's declination and
latitude of place.

Ans. Lat., 29° 58' 39".

Dec., 4° 58' 39" S. in S. lat.
or N. in N. lat.

11. The altitude of Sirius on the prime vertical is found to read 39° 48'.
The declination of Sirius is 16° 35' 20" S. Find the latitude of the observing
station. Allow for refraction.

Ans. Lat., 26° 30' 1" S.

THE DETERMINATION OF LATITUDE. 179

12. At a place in South latitude the altitude of a star was observed at
its upper and at its lower culminations, the altitude corrected for refraction
at upper culmination being 60° 45' 15". and at lower culmination 10° 16' 15".

Find the latitude of the place of observation and the declination of the

star.

Ans. Lat., 35° 30' 45".

Dec. S., 64° 45' 30".

13. On the evening of 8th February, 1914, at a place in S. latitude, the
magnetic bearing of ft Hydri at its Western elongation was 185° 47' 35",
and that of 0 Argus a* its Eastern elongation was 137° 24' 42".

Declination of ft Hydri, .... 77^44'29"S.
0 Argus, .... 63° 56' 36" S.
Determine the latitude of the place and the magnetic variation.

Ana. Latitude, 36° 24' 56".
Variation, 9° 30' 20" E.

14. The altitude of Regulus at 10 hrs. 08 min. sidereal time was 46° 52' 32"
(fully corrected). From the Nautical Almanac we find : —

R. A. of Regulus, 10 hrs. 03 min. 17 sec.

Declination of Regulus, . . . .12° 26' N.
What was the correct altitude when on the meridian ?

Ana. 46° 52' 37-4".

15. On 9th March, 1914, at a place South of Equator in 140° E. longitude
the following altitudes of a Virginis (Spica) were observed near its meridian
passage and their times taken with a chronometer keeping local mean
time : —

Observed Altitudes. Local Mean Times.

57° 40' 36", 2 hrs. 02 min. 18 sec. a.m.

44' 34", ..... 05 min. 54 sec. „

48' 40", ..... 10 min. 50 sec. „

50' 10", ..... 15 min. 58 sec. ,,

49' 30", 22 min. 10 sec. „

46' 40", 27 min. 00 sec. „

42' 35". ..... 31 min. 02 sec. „

The sidereal time at G.M.N., March 8th,,is 23 hrs. 1 min. 22-91 sec.
R.A. of Spica = 13 hrs. 20 min. 41-4 sec.

Declination of Spica = 10° 43' 00" S.

Find the latitude of the place.

Ans. 42° 52' 51".

16. The declination of a star being 40° S., what are the latitudes of the
places where its meridian altitude will be 80° ?

Ans. 50° or 30° S.

180 ASTRONOMY FOR SURVEYORS.

17. In south latitude two stars are observed on the meridian, one north
and the other south of the zenith, the difference of zenith distances being
found to be 13' 03-45" N., the declinations of the stars being 45° 38' 37-48" S.
and 42° 44' 04-63" S. respectively.

Find the latitude.

An*. 44° 17' 52-8".

18. A south circumpolar star was observed at equal intervals shortly
before and after its elongation, when it was found to change its altitude
from 44° 35' to 47° 35', during an interval of 19 min. 47 sec., by watch
keeping correct mean time.

Find the polar distance of the star and the latitude of the place of
observation.

Ans. 37° 20' 30".

Latitude = 33° 27' 58.

19. At 6.10 p.m., local mean time, by watch on 15th September, 1907^
in longitude 151° 06' 30" East, the magnetic bearing of r Octantis was
170° 37' 30", the bearing of the referring mark being 72° 50' 45", and the
observed altitude of the star was 34° 36'.

R.A. of Octantis, . . . .19 hrs. 12 min. 48 sec.
Declination of Octantis, . . 89° 14' 49" S.

Sidereal time at G.M.N., Sept. 15th, 11 hrs. 33 min. 12 sec.
Sept. 14th, 11 hrs. 29 min. 15 sec.

Find the latitude of the observer and the true bearing of the referring
mark.

Ans. Latitude = 33° 54' 19".

20. On March 6th, 1914, the altitude of Polaris, when corrected for
instrumental errors and refraction, is found to be 46° 17' 28", the mean
time of observation being 7 hrs. 43 min. 35 sec. p.m. and the longitude of
the place 37° W.

Sidereal time at G.M.N., March 6th, 22 hrs. 53 min. 29-8 sec.

R.A. of Polaris, March 6th, . . 1 hr. 27 min. 37-3 sec.

N. declination of Polaris, March 6th, 88° 51' 8"
Find the latitude.

Ans. N. 46° 3' 35".

21. The observatory at Stockholm is in latitude 59° 20' 33" N., and that
at the Cape of Good Hope in latitude 33° 56' 3-5" S. The declination of
Sirius is 16° 35' 22" S. Find the altitudes of Sirius when on the meridian
at Stockholm and at the Cape of Good Hope respectively.

Ans. 14° 04' 05" and
72° 39' 18-5".

22. The upper transit of a South circumpolar star was observed to occur
at 7 hrs. 05 min. 28 sec. p.m. local mean time, and to reach its greatest

THE DETERMINATION OF LATITUDE. 181

western elongation at 11 hrs. 44 min. 30 sec. p.m., when its observed azimuth
was 33° 48'.

Find the latitude of the place of observation and the declination of the
star. , Ans. Latitude, 31° 02' 52" S.

Declination, 61° 32' 11" S.

23. On March 13th, 1911, at a place South of the Equator, in longitude
9} hours E., at 6 minutes before apparent noon, the altitude of the sun's
lower limb was found to be 58° 04' 20", at which time clouds prevented
further observation. The sun's declination at G.M.N., March 13th, is
3° 15' 07-4" S., and on March 12th 3° 38' 41-8" S.

Find the latitude of the place by reduction to the meridian, the sun's
.semi-diameter being 16' 07", its parallax 5", and refraction 37".

Ans. 35° 02' 28".

24. The altitudes of a star when it crosses the meridian and the prim©
vertical of a place are a and b. If Hs the latitude of the place, show that

cot I = tan a — sec a sin b.

25. The meridian altitude of Altair is 51° 55' 45", its declination being
8° 34' 34" N. and the meridian altitude of 3 Pavonis is 52° 54' 32", its North
polar distance being 156° 36' 18". Find the latitude of the place of obser-
vation.

Ans. 29° 30' 15-5" S.

26. At a place, south of the equator, the meridian zenith distances of the
two stars y* Norma and <r Scorpii were observed, the former to the south,
the latter towards the north. The observed difference of the zenith distances
was found to be 19' 21". Find the latitude of the place of observation.

Declination of y2 Norma, . . 49° 57' 08-3" South

ff Scorpii, . . 25° 23' 31-2" South

Another observer, stationed some distance to the north, found the differ-
ence of the zenith distances of these stars to be exactly the same. Deter-
mine his latitude also.

Ans. 37° 50' 00-25" and
37° 30' 39-25".

27. The mean altitude reading from four observations of Polaris was
51° 39' 34-25", the mean readings of the alidade level E., 5-5, 0., 6-5, one
division of level = 15", mean chronometer time 7 hrs. 09 min. 54-8 sec.,
the chronometer keeping L.M.T. and being 3 min. 24 sec. fast. The longi-
tude of station was 0 hr. 2 min. 9 sec. E. G.S.T. at G.M.N. on the day of
observation was 13 hrs. 05 min. 34-1 sec. Declination of Polaris, 88° 45'
50-8" ; R.A. of Polaris, 1 hr. 22 min. 26 sec. Barometer, 30-27". Ther-
mometer, 42°. Compute latitude of place. (Example from " Topographical
Surveying," by Major Close.)

Ans. 51° 23' 34".

182

CHAPTER X.

THE DETERMINATION OF TIME BY OBSERVATION.

IN this chapter it is proposed to consider the principal
methods available to the surveyor for the practical
determination of the local mean or sidereal time by
observation. Other methods have been devised, but
the methods about to be described are those that have
proved in practice to be the most convenient and satis-
factory. Nearly all the ordinary time determinations of
the surveyor are made by the second of the following
methods, a convenient observation that may be carried
out in the day light, and by which the time may be
readily found with ordinary instruments with an error
of not more than one or two seconds. One second of time
will, of course, correspond to 15" of hour angle.

First Method — By Meridian Transits. — We know that the
local sidereal time at the instant that a star is on the
meridian is measured by the R.A. of the star. Conse-
quently, if we make the observation upon a star whose
R.A. is known, by setting a theodolite up in the meridian
and noting the time of transit of the star across the
vertical wire, we have clearly a very simple way of finding
the sidereal time at that instant and thus of determining
the error of a watch or chronometer.

A similar observation may be made upon the sun, by
noting the times of transit of the E. and W. limbs. The
mean of these times will be the time of transit of the sun's
centre, which takes place at apparent noon. From the
Nautical Almanac we can find the equation of time for
the given date, from which the mean time at the instant

DETERMINATION OF TIME BY OBSERVATION. 183

may be found. If only one limb be observed, then allow-
ance must be made for the time occupied by the sun's
semi-diameter in crossing the meridian, which is given
in the Nautical Almanac on page 1 for each month.

EXAMPLE. — On December 1st, 1914, at a place in longitude 9 hrs. 45 min. E.,
the meridian times of transit of the E. and W. limbs of the sun across the vertical
wire of a theodolite were taken with a watch supposed to keep the standard time
of the meridian 9 hrs. 30 min. E. The observed times of transit being 11 hrs.
32 min. 32-5 sec. and 11 hrs. 34 min. 52-5 sec., determine the error of the
watch.

From the Nautical Almanac we find that at Greenwich apparent noon
on December 1st, 1914, the equation of time, to be subtracted from apparent
time, is 11 min. 6-47 sec., and that it is decreasing, the variation in 1 hour
being 0-918 second.

Therefore, 9| hours before this — i.e., at apparent noon in longitude
9 hrs. 45 min. E. — the equation of time will be 11 min. 6-47 sec. + 9|
X 0-918 sec. = 11 min. 15-4 sec.

.-. L.M.T. at L.A.N. - 11 hrs. 48 min. 44-6 sec.

.-. Standard time at L.A.N. = 11 hrs. 33 min. 44-6 sec.

But the time of transit of the sun's centre — i.e., the mean of the two
observed times — was 11 hrs. 33 min. 42-5 sec.

Therefore, the watch was 2 seconds slow.

Fig. 42.

The Effect of an Error in the Direction of the Meridian.
—If the instrument be in accurate adjustment, but the
direction of the meridian be in error, then the meridian
set out will pass through the zenith of the observer, but
not through the celestial pole. In Fig. 42, let Z C denote

184 ASTRONOMY FOR SURVEYORS.

the erroneous meridian, making an angle that we will
call e with the true meridian Z P A. Then a star will
intersect the apparent meridian at S, and the time noted
will be either too soon or too late, according as the meridian
is wrongly marked out to the East or West of the true
direction, the error being measured by the hour angle
S P Z, which we will call /L

P Z = c = co- latitude
P S= p= polar distance of star.
Then, in the triangle P Z S,

cot p sin c = cot e sin h -j- cos c cos h.

Since e and h are both small, we may write, without
appreciable error, h and e instead of sin h and sin e respec-
tively, and may put cos h and cos e each = 1.

. . e (cot p sin c — cos c)= h.

A= a85?. <«-.?). (1)

sin p

Thus h will have its smallest value when p is nearly = c ;
that is to say, when the observed star makes its meridian
transit near the zenith.

If in equation (1) c== 60°, or the latitude of the place
is 30°, and p= 40°, then, if e= 01' of arc, h= 32" of arc
or 2 seconds of time. Thus, in this case, an error of
1 minute of arc in the direction of the meridian will
make the time of transit wrong by two seconds.

It is clear, therefore, that the method requires the
meridian to be very accurately set out, and the instrument
must be in perfect adjustment, if good results are to be
obtained by this method.

In Fig. 42 we have illustrated the case where the star
transits above the celestial pole. If the lower transit
had been observed, then the angle h would be the supple-
ment of the angle S P Z, and in this case the formula

DETERMINATION OF TIME BY OBSERVATION. 185

sin (c + p)

would become h = e — .

sin p

Both are included in the general formula,

sin zenith distance cos alt.

h= e —, or e

sin p cos dec.'

\\hich applies to all cases.

The error is thus very great if the polar distance of the
star is small, and is least for those stars that transit
near the zenith.

Fig. 42a.

The Effect of an Error in the Horizontality of the Trans-
verse Axis. - - The direction of the meridian may be
accurately set out with the telescope horizontal or nearly
so, and yet, if the transverse axis is not horizontal, the
line of sight may depart considerably from the meridian
at high altitudes. If the angle made by the transverse
axis with the horizontal be determined by means of the
striding level, the necessary correction to the time of
transit may be made as follows :—

In Fig. 42a, the meridian actually swept out by a
telescope with the transverse axis slightly tilted is repre-
sented by A S B, A and B being the North and South
Points, and Z the zenith. The transit of the star is
observed in consequence at a point S on this circle, and
the error in time is measured by the angle S P Z.

186 ASTRONOMY FOR SURVEYORS.

In the triangle EPS

S P = p = polar distance of star,
B P= 180°— 1= supplement of latitude,
Angle P B S — e = error measured by striding level,
Angle B P S= x= required error in time of transit.

.-. cot S P sin B P = cot e sin x -f- cos B P co^ x.
.-. treating x and e as small quantities,

x
cot p sin / = - - cos I.

sin (Z-f p) sin altitude

x=e - — , ore-

sin p cos dec.

This formula gives us the hour angle of the star at the
moment of observation. Usually e and, therefore, x will
be in seconds of arc, and x must then be divided by
15 to determine the error of the observed time of transit
in seconds of time. Clearly the transit will be observed
either too soon or too late according to the direction of
tilt of the transverse axis.

If the star transits below the pole, x will be the supple-
ment of the angle B P S, and we get

sin (I— p) sin alt.

x= e— - •, which again = e -

sin p cos dec.

The error in time in this case increases with the altitude.

EXAMPLE. — At a place in latitude 30° S. the sidereal time of transit
of a star across the meridian is observed to be 12 hrs. 30 min. 17-5 sec., the
declination of the star being 58° 30' S. The readings of the striding level,
one division of which = 13", are : —

L. R,

6-0 5-0

3-6 . 7-2

9-6 12-2

9-6

4 ) 2-6

0-65
0-65 X 13 --= 8-45".

DETERMINATION OF TIME BY OBSERVATION 187

sin 61° 30'

.-. error m hour angle = 8-4o X — — ^rs-^7 = 14-21 .

sin 31° 30

This is equivalent to 0-95 second of time.

As the right-hand side of the axis is the higher, and the telescope is directed
towards the South, the transit is, therefore, observed too soon by this amount-,
and the corrected time of transit across the meridian is 12 hrs. 30 min.
18-45 sec.

Meridian Transits on Both Sides of the Zenith.— A consider-
able improvement may be made in the accuracy of the
method by taking observations of the times of transit
of two stars, one on each side of the observer's zenith.

In Fig. 43, let Z denote the zenith, P the celestial pole,
A Z P B the direction of the true meridian, and C Z D
the direction of the meridian actually set out, the figure
being drawn as though the celestial sphere were viewed

Fig. 43.

from above. Suppose that the times of transit of two
stars are observed, one at St and the other on the opposite
side of the zenith as at S2. Then, since both stars move in
the same direction, as shown by the arrows, if the observed
time of transit of Sj is later than it should be, owing to
the faulty determination of the meridian, the time of
transit of S2 will be correspondingly earlier. If the stars
are well selected, it may be that the time errors of the two
observations are equal and opposite, so that the mean of
the two results will give a correct time determination in
spite of the error in the setting out of the meridian. This
will be the case if the hour angle Sx P Z is = the angle
S2 P Z, for then one observation will be just as much
too soon as the other one is too late.

188

ASTRONOMY FOR SURVEYORS.

The conditions that this may be the case are readily
obtained as follows : —

Let angle B Z D = e= meridian error, and suppose that
the hour angle Sx P Z = S2 P Z == h,

c = co-latitude P Z.

Then, from the triangles Sx P Z, S2 P Z,
% sin h sin Z Sx sin Z S2
sin e sin P Sx sin P S2

But, since the error e is small, we may write very
approximately P Sj = c — Z Sj^ and P S2 = c + Z S2.
sin (c - Z SJ _ sin (c + Z S2)

sin Z Sj sin Z S2

sin c . cot Z Sl— cos c= sin c cot Z S2 + cos c.

cot Z Sj — cot Z S2 = 2 cot c.

This, then, is the condition that has to be satisfied
by the zenith distance of the two stars if the observations
are to be so balanced that by taking the mean of the
two we eliminate, or nearly so, the error due to a faulty
setting out of the meridian.

The following table, based upon the above formula, gives
the proper zenith distance of the star on the opposite side of
the zenith to the pole, corresponding to different zenith dis-
tances of the other observed star, for different latitudes :—

Zenith
Distance
of Star

Zenith Distance

of Star o
to the

i Opposit
Pole.

e Side of Zenith

Side as

Pole.

Lat. 10°.

Lat. 20°. Lat. 30°.

Lat. 40°.

Lat. 50°.

Lat. 00°. Lat. 70°. Lnt. 80°.

5°

5° 09'

5° 20' 5° 34'

5° 51'

6° 18'

7° 09' 9° 34' 85° 0'

10°

10° 39'

11° 26' 12° 29'

14° 30'

16° 55'

24° 22' 80° 0'

20°

22° 40'

26° 21' 32° 08'

43° 05'

70° 0'

30°

35° 56'

44° 53' 60° 0'

86° 55'

40°

50° 0'

65° 07' 87° 53'

. .

50°

64° 04'

83° 39'

•

60°

77° 20'

70°

89° 21'

DETERMINATION OF TIME BY OBSERVATION. 189

The advantage of selecting the two stars in this way
may be illustrated by a computed example. Suppose
that the place of observation is in latitude 30°, and that
the polar distance of the 'star observed on the same
side of the zenith as the pole is 40°, so that its zenith
distance is about 20°. Suppose, further, that the marked
meridian is as much as 1° in error.

Computing with these data the spherical triangle
S P Z of Fig. 42, it may be shown that the hour
angle S P Z is 2 min. 04-8 sec. In other words,
the observed transit will take place too soon by thi&
amount.

Now, according to the table, the star observed on the
opposite side of the zenith should have a zenith distance
of 32° 08'. Suppose it actually has a zenith distance of
32°, equivalent to a polar distance of 92°. Then, computing
in the same way the hour angle of this star when on the
faulty meridian, we find that its observed transit will
be too late by 2 min. 04 sec.

Thus from one observation the chronometer would be
set too fast by 2 min. 04 sec., and from the other it would
be set too slow by about the same amount, and the mean
of the two observations would give the time correct to
the nearest second in spite of the fact that the direction
of the meridian is 1° in error.

If, however, the zenith distances of the two stars are
not balanced in the way indicated, the accuracy of the
mean result is nothing like so great. If, for example, the
two zenith distances were the same, the star observed
on the opposite side of the zenith to the pole having a
zenith distance of 20°, or a polar distance of 80°. Then,
on computing the spherical triangle, it will be found
that the observed transit of this star is too late
by 1 min. 24 sec., so that the mean of the two
observations is then in error to the extent of about
20 seconds.

190

ASTRONOMY FOR SURVEYORS.

Second Method — By Extra Meridian Observations of Sun or
Star. — This is, as a rule, the most convenient and suitable
method for the determination of time by the surveyor.
It consists in the measurement of the altitude of sun or
star when out of the meridian, at the same instant noting
the chronometer time. Then, from a knowledge of the
latitude of the place and the declination of the body
observed we may compute the proper local time at the
instant of observation, and so determine the error of the
chronometer.

The most favourable time for making such an obser-
vation will be when the altitude of the celestial body is

Fig. 44.

changing most rapidly, and this will be the case when
it is near the prime vertical. This position has also
other advantages, as we shall see in the course of the
discussion.

As an altitude has to be measured, refraction must be
allowed for, and as there is considerable uncertainty
about this at low altitudes, the star observed should have
an altitude of at least 15°.

The method involves the solution of the same spherical
triangle that we have discussed in connection with extra-
meridian observations for azimuth. Thus, in Fig. 44, if

DETERMINATION OF TIME BY OBSERVATION. 191

S is the star observed, then in the spherical triangle
Z P S we know the three sides :—

Z P = c = co-latitude,
S P = p = polar distance of star,
Z S= z= zenith distance, or the complement of
the observed altitude.

Therefore, we can compute the hour angle S P Z, from
which we can find the local sidereal time if we know the
R.A. of the star, or this at once gives us the local apparent
time in the case of the sun.

Let the angle S P Z =h.

Then, we have three available formulae adapted to
logarithmic computation, any one of which may be used
for computing h. They are —

if s = ^ (z + c -f- p)

h /sin (s — c) . sin (s — p)
sin - = y

2 sin c . sin p

h /sin s . sin (s — z)
cos -- = v — »

2 sin c . sin p

h /sin (s— c) . sin (s — p)

tan - = y - ~ : — ; : — -

2 sin s . sin (s — z)

The Choice of a Formula. — Of the three formulae, that
for cos is somewhat the simplest, as we must find s in

any case, and we have then only to find s — z in addition.
With the sine formula we have one more subtraction to
make, but there is the advantage that only tables of log
sines are used, and there is less risk of mistake in taking
out the logarithms.

If, however, we are utilising the same observation, as
may be done, for the determination of azimuth in addi-
tion, then we shall require to compute also the angle
S Z P. In this case it is a decided advantage to select

192 ASTRONOMY FOR SURVEYORS.

the tangent formula for the computation of both angles,
for we shall then need only to look up four logarithms,
as the same expressions sin s, sin (s— c), sin (s— p), and
sin (s — z) will occur in the tangent formulae for both
angles. If, on the other hand, we use the sine or cosine
formula for the two angles, it will be necessary to look
up six logarithms.

Another important point in the selection of a formula
is this. The variation in value of the tangent of an angle,
as the angle increases from 0° to 90°, is very much greater
than in the case of a sine or cosine. Consequently a table
of tangents will enable us to determine the value of an
angle with greater precision than a table of sines or
cosines. This is of practical importance when the angle
under consideration is near to 0° or 90°. Thus there is
very little variation in the value of the cosine of an angle
up to 2° or 3°, and, if we wish to determine the values
of such small angles to seconds, a table of cosines is not
nearly so good as a table of tangents. Similarly, there
is very little variation in the sine of an angle near to 90°,
and it becomes difficult to compute such angles with
precision from a sine table. It follows, therefore, that
if h is near 0° or near to 90°, the tangent formula is the
best one to adopt.

Data Necessary for Computation. — In addition to the
measured altitude, we require a knowledge of the latitude
of the place and the declination of the body observed.
The declination for a star is taken straight from the
Nautical Almanac, but the declination of the sun has to
be found by using approximate values for the longitude
and local time. If the result obtained shows that the
assumed local time is -very much out, the calculation
should be repeated by using the corrected value of the
local time found from the first computation.

Arrangement of the Computation.— It is worth some trouble
to make a neat form for the computation. A good

DETERMINATION OF TIME BY OBSERVATION. 193

arrangement reduces the work, and is an aid to accuracy.
The following, for instance, is the method adopted in the
printed forms of the Queensland Survey Department :—

p = 59°34/48// log sin 9-9356770

c= 76° 05' log sin 9-9870611

z= 66° 34' 19" 19-9227381

2 ) 202° 14'0r' subtract from 20-

9 n_ 41° 32' 15-5" loS~ -- : — = 0-0772619

s ~ P ~ sin p sin c

8-c = 25° 02' 03-5- iogsin 9-8215856

log sin = 9-6265032

2 ) 19^5253507

.-. i h= 35° 22' 48" log sin 9-7626753

Where the same observation is to be utilised for both
time and azimuth, a neat device is to proceed as follows : —

log sin (s— c) = say 9-949960 '
log sin (s—p) = 9-046045

Iogsin (s— z) 9-875721

28-871726

subtract log sin s 9-945558

2 ) 18-926168

9-463084

From this we have simply to subtract log sin (s — z)

7 ry

and log sin (s — p) in order to get tan and tan -f
respectively.

9-463084 9-463084

log sin (s—z) = 9-875721 9-046045

log tan- 9-587363 log tan - 10-417039

Having Computed the Hour Angle to Find the Time of
the Observation. — In the case of a star the angle S P Z,
turned into time by dividing by 15, measures the interval

194 ASTRONOMY- FOR SURVEYORS.

of sidereal time after or before the time of culmination,
according as the star is observed on the West or East
of the meridian. But the R.A. of the star is equal to the
sidereal time at the instant of culmination. Therefore,
the sidereal time at the moment of observation is obtained
by adding (or subtracting) the value of h to the R.A. of
the star. This may be turned into mean time in the way
already discussed.

Thus, if the R.A. of the star is 7 hrs. 30 min., and the
angle h is 35°, the star being observed in the West, then
the local sidereal time at the moment of observation is
7 hrs. 30 min. + 2 hrs. 20 min. = 9 hrs. 50 min.

If the sun has been observed, the value of the angle
h at once gives us the interval of solar time before or after
the meridian transit of the sun — that is to say, it gives
us the local apparent time. To convert this into mean
time the equation of time must be determined at that par-
ticular instant. To do this we first find the corresponding
Greenwich apparent time, by allowing for the difference of
longitude, and then take the equation of time from page 1
of the Nautical Almanac, allowing for the hourly variation.

Suppose, for example, that the angle k, for a sun observation, is 48° 20',
the observation being made at a place in longitude 60° W. on May 23rd in
the afternoon. We have, therefore,

Local apparent time, . . .3 hrs. 13 min. 20 sec.
Longitude, . . . . .4 hrs. 0 min. 0 sec.

Greenwich apparent time, May 23rd, 7 hrs. 13 min. 20 sec.

We have then to find the equation of time at this instant. The Nautical
Almanac gives for this date, 1914, the equation of time at apparent noon,
Greenwich, as 3 min. 30-40 sec. The variation in one hour is given as
0-191 second, the equation decreasing on successive days. The Almanac
states that the equation of time is to be subtracted from apparent time.
Hence, at the given instant,
Equation of time = 3 min. 30-40 sec. — 7-222 x 0-191 sec. = 3 min.

29-02 sec.

Therefore, the required mean time is
3 hrs. 13 min. 20 sec. — 3 min. 29-02 sec. = 3 hrs. 09 min. 50-98 sec.

DETERMINATION OF TIME BY OBSERVATION. 195

Averaging Several Observations of the Same Star. — In
practice it is usual to take at least two, and commonly
four, observations in as quick succession as possible, half
being taken with F.L. and half with F.R. The computa-
tion is then made as though one observation only had
been taken, the mean of the altitudes being assumed
to be the true altitude at the mean of the noted chrono-
meter times.

The object of this procedure is to eliminate instrumental
errors, but this is done at the expense of introducing
another error due to the fact that the assumption made
is not mathematically exact. The investigation of the
magnitude of the error thus introduced into the work is
too complex for insertion here, but it may be stated that
the surveyor is quite safe in thus averaging altitude
observations extending over a range of 2° in altitude
under ordinary conditions. The error thus made in an
extra- meridian time determination is then generally only
a small fraction of a second of time, its exact magnitude
depending upon the latitude of the observer, the declina-
tion, and hour angle of the heavenly body. It is least
when the hour angle is nearly 90°.

Observations on Both East and West Stars. — It is a great
improvement in accuracy to take one set of observations
upon a star in the east and another corresponding set,
under as similar conditions as possible, upon a star in
the West. The averaging of two such sets of observations
tends to eliminate certain classes of errors, and this should
always be done where the highest accuracy is sought.
If, for example, the refraction assumed is too great, the
corrected altitude will be too low, and the computed time
will be too early for a star in the east, while it will be
correspondingly too late for a star in the west. If the
two errors are about equal, as will be the case if the E.
and W. stars make about the same horizontal angle
with the meridian, and are observed at about the same

196

ASTRONOMY FOR SURVEYORS.

altitude, then the average of the two sets of results will
be correct. Similarly, the effects of any systematic
error in the measurement of altitude are eliminated by
pairing sets of observations in this way. The same
applies to extra meridian observations for azimuth.

EXAMPLE OF EXTRA MERIDIAN OBSERVATION ON SUN FOR TIME.

Forenoon Observations.

Place — Survey Office, Adelaide. Thermometer — 56°.

Longitude — 9 hrs. 14 min. 20 sec. E. Barometer — 30-49 inches.
Latitude— 34° 55' 38" S. Date— -15th July, 1914.

Value of 1 division of bubble — 10". Standard Meridian — 9 hrs.

30 min. E.
Chronometer keeping approximately standard time.

OBSERVATIONS.

Vertical Angles.

Level.

Observed
Limb.

Face.

Chronometer
Time.

B. Mean.

H. M. S.

20° 08' 50"

20° 09' 10" 20° 09' 00" 10

9 30 23

R 20° 49' 10"

20° 49' 00 '' 20° 49' 05" 11 9

9 31 29

R 20° 27' 20"

20° 27' 00" 20° 27' 10" 11-5 8-5

9 32 40

L 21° 07' 20"

21° 07' 50" j 21° 07' 35" : 10 1 10

9 33 44

Means, . ' 20° 38' 12"

10-6

9-4

9 32 04

Computation for sun's declination at assumed approximate time of
observation.

Approximate standard time of observation,

14/7/14, .
Difference for standard meridian,

Corresponding G.M.T., .

Declination : 14th July, 1914 (G.M.N.), .
Difference for 12 hrs. 02 min. 04 sec.,

21 hrs. 32 min. 04 sec.
9 hrs. 30 min. 00 sec.

12 hrs. 02 min. 04 sec.
—

21° 47' 03-3"
04' 28-6"

Declination at instant of observation (North), 21° 42' 34-7"
Sun's South Polar Distance, . . 111° 42' 34-7"

DETERMINATION OF TIME BY OBSERVATION. 197

h I sin (s — c) sin (s — p)

Formula— Tan - = / — '— — * — —£L.

'Y sin s sm (a — z)

CALCULATION.

Mean of observed altitudes, . . . 20° 38' 12"
Level correction, ..... 6"

20° 38' 06"
Refraction and parallax, .... 2' 21"

Corrected altitude, . . . .20° 35' 45"

Zenith distance = z, .... 69° 24' 15"

Co-latitude = c, 55° 04' 22"

Sun's polar distance = p, . . . 111° 42' 35"

2s, 236° 11' 12"

*, 118° 05' 36"

s-c, 63° 01' 14"

s - p, 6° 23' 01"

s-z, 48° 41' 21"

log sin (s-c), 9-949960

log sin (s - p), ... . 9-046045

logcosecs, .... . 10-054442

log cosec (.9 - z), 10-124279

log tan"*, 19-174726

log tan | = tan 21° 08' 28", . - 9-587363

h, 42° 16' 56"

h (in time), ...... 2 hrs. 49 min. 08 sec.

Local apparent time = 24 hrs. — h, . .21 hrs. 10 min. 52 sec.
Longitude, , . . . . . . 9 hrs. 14 min. 20 sec.

Greenwich apparent time, . . . 11 hrs. 56 min. 32 sec.

198 ASTRONOMY FOR SURVEYORS

Equation time at G.A.N., ... 5 min. 33 sec.

Correction for 11 hrs. 56 min. 32 sec., . 3 sec.

Equation time instant observation, . 5 min. 36 sec.

L.A.T., . .... 21 hrs. 10 min. 52 sec.

L.M.T., . ...... . .. . 21 hrs. 16 min. 28 sec.

Diff. Standard Merid., . 15 min. 40 sec.

Local Standard time, . . . . 21 hrs. 32 min. 08 sec.
Chronometer time, . . .-."• . . 21 hrs. 32 min. 04 sec.

Error of Chronometer, . . * • 04 sec. slow

EXAMPLE FOR REDUCTION.

With the same instrument as that used in the preceding observation
a similar set of four sun observations was taken on the afternoon of July
21st, 1914, at the same place. The mean altitude obtained was 23° 53' 36",
the average alidade level readings were E. 10-5, 0, 9-5. The mean of the
chronometer times was 2 hrs. 52 min. 52-5 sec.
From the Nautical Almanac —

Declination of sun, at G.M.N., July 20th, 1914, 20° 47' 18-2" N.
Variation in one hour at noon on the 20th, . 27-60"

21st, . 28-47"

Equation of Time, G.A.N., July 20th (to be added

to apparent time), . . . . . 6 min. 05-99 sec.

Variation in one hour, . ? . . . 0-165 sec.

Longitude, standard time, and latitude are given in the preceding case.

The chronometer being supposed to keep standard time, determine its

error. Ans. 02-1 sec. slow.

The Effect of an Error in Latitude. — It is important that
we should know to what degree of precision the latitude
must be known in order that the time may be determined.
This may be readily investigated in a manner similar to
that adopted with corresponding problems previously.

From the spherical triangle S Z P of Fig. 44,
cos z = cos c cos p-\- sin c sin p cos h.

If c is too large by a small amount y, then, for the
same measured zenith distance z, h will be too small by
an amount x, and we shall have

cos z = cos (c + y) cos p-\- sin (c + y) sin p cos (h — x).

DETERMINATION OF TIME BY OBSERVATION. 199

Subtracting these two equations, and treating x and h
as small quantities, we readily get

— cos c cos h sin p -}- sin c cos p
x=y

cot Z

sin c

sin c sin p sin h

where Z denotes the azimuth angle S Z P.

This shows that x will be very large compared with y,
if Z is nearly equal to 0, or if c is nearly 0. That is to
say, a small error in the latitude will produce a very large
error in the time if the body is observed near to the
meridian, or if the observation is made in high latitudes
near to either terrestrial pole.

On the other hand, if Z is 90° — i.e., if the observation is
made on the prime vertical — x is 0, and an error in latitude
makes no difference. In this case the angle S Z P is a right-
angled triangle, and we can get a relation between p, z,
and h that does not involve c at all, so that a knowledge
of the latitude is unnecessary. If the observation is made
near to the prime vertical, therefore, an error in latitude
will produce very little effect on the time determination.

The following table, based upon the above formula,
gives the error in time corresponding to an error of 1'
in the latitude for different azimuth angles :—

ERROR IN TIME CORRESPONDING TO 1' ERROR IN LATITUDE.*

Azimuth of
Observed Body.

Latitude of Place.

0°.

30°.

40°.

50°.

60°.

Seconds.

* Seconds. Seconds.

45°

4-0

4-6

5-2

6-2

8-0

60°

2-3

2-6

3-0

3-5

4-5

80°

0-7

0-8

0-9

1-1

1-4

90°

0-0

* If the word Declination be substituted for latitude, the same table will
give the error in time due to an error of 1' in the Declination, the first column
representing, not the azimuth, but the angle Z S P.

200

ASTRONOMY FOR SURVEYORS.

This all points to the desirableness of making the
observation as near to the prime vertical as possible.

The Effect of an Error in the Measured Altitude. — By
a method similar to that adopted in the last paragraph it
may be readily shown, if x is the error in the hour angle
corresponding to an error y in the observed altitude, that

x = y cosec Z cosec c

x clearly becomes very great if either Z or c are small,
and it has its least value when Z and c are each 90°. Thus,
again, an error of observation has the least effect when
the observation is made on a celestial body near the
prime vertical, and the most favourable place for making
the observation is at the equator.

TABLE SHOWING ERROR IN TIME DETERMINATION OWING TO AN ERROR
OF 1' IN THE MEASURED ALTITUDE, WITH DIFFERENT AZIMUTHS OF
THE OBSERVED BODY.

Latitude of Place.

Azimuth of

Observed Body.

0°.

30°.

40°.

50°.

60°.

Seconds.

Seconds

Seconds.

45°

5-6

6-4

7-3

8-7 11-3

60°

4-6

5-3

6-0

7-1 9-2

80°

4-1

4-7

5-3

6-3

8-1

90°

4-0

4-6

5-2

6-2

8-0

This table deserves a little careful consideration, as it
shows the degree of precision with which altitudes must
be measured if the time is to be determined within one
second. Under the most favourable possible conditions
an error of J minute of arc will cause an error of one
second in the time, and it may produce an error of two
seconds or even more.

EXAMPLE. — In the extra -meridian observation for time set out at length
in paragraph just preceding show that an error of 1' in the measured
altitude will produce an error of 7 seconds in the time.

DETERMINATION OF TIME BY OBSERVATION. 201

The Effect of an Error in the Declination of the Sun caused
by a Defective Knowledge of Longitude or Local Time. — With
star observations the Nautical Almanac gives us the
declination of the star with all the precision that is re-
quired, but with sun observations the surveyor has first
of all to compute the declination. To do this he requires
to know both his longitude and the approximate local
mean time.

From the formula

cos z = cos c cos p + sin c sin p cos h

it appears that the relation between an error in p and an
error in h will be of precisely the same nature as the
relation between an error in c and an error in h. So that
if x denotes the error in the hour angle corresponding to
an error y in the declination

cot Z S P

x= -•—— . y.
sin p

Thus the table already given, showing the error in time
caused by 1' error in latitude, also gives the error in time
caused by 1' error in declination, provided that the
first column is taken as representing the angle Z S P
instead of the azimuth.

We have already seen that the maximum rate of varia-
tion of the declination of the sun is a little less than 1'
per hour. So that to get the declination of the sun to
the nearest minute it is sufficient to know the time to the
nearest hour. But one hour of time corresponds to 15°
of longitude, so that it is seldom that the surveyor will
not know his longitude sufficiently well for this purpose.

It will be seen from the table that, in order to deter-
mine the time to the nearest second, it will be necessary
to know the declination within only about one-fifth of a
minute of arc under almost the worst conditions of obser-
vation considered in the table. For this it will be usually
sufficient to know the local time within a quarter of an hour.

202

ASTRONOMY FOR SURVEYORS.

If the local time is not known with sufficient accuracy,
its value must be assumed for the purpose of finding
the approximate declination. This is then used in a
preliminary calculation made to determine the time.
The calculation is then made over again, using the approxi-
mate local time so found in order to get a more accurate
value of the sun's declination, which in turn is used in
the computation to obtain a more accurate determination
of the local mean time.

Third Method — By Equal Altitudes. — If a star be observed
at the same altitude on opposite sides of the meridian ,
the two observations must clearly be made at equal
intervals of time before and after the star's meridian

Fig. 45.

transit. Thus, in Fig. 45, if the star be observed in the
two positions, Sj and S2, so that the zenith distances-
Z Sj and Z S2 are equal, then, if P is the celestial pole,
the two hour angles Z P Sx and Z P S2 must be equal.

It follows that the mean of these two observed times
is the time of the star's meridian transit. But the local
sidereal time at the instant of the star's meridian transit
is determined by the star's R.A., which is given by the
Nautical Almanac. This local sidereal time may be
reduced to mean time, and a comparison of this with
the average of the two observed chronometer tinier
determines the error of the chronometer.

DETERMINATION OF TIME BY OBSERVATION. 203

With stars the method is capable of giving very accurate
results, and it has the great advantage that no knowledge
is required of latitude, declination, or even azimuth, and
errors of graduation of the instrument have no effect
upon the result. But to the surveyor it has the obvious
drawback that a considerable interval of time must
elapse between the observations.

As the accuracy of the determination depends upon
the altitude being the same at the two observations, the
star should have an altitude of something more than 45°,
in order to get rid of the uncertainties of refraction near
the horizon.

EXAMPLE. — On September 1st, 1914, jj Crucis was observed East of the
meridian at 10 hrs. 42 min. 30-5 sec. by a chronometer keeping sidereal
time. It was again at the same altitude West of the meridian at 14 hrs.
51 min. 20-7 sec. Find the error of the clock.

East 10 hrs. 42 min. 30-5 sec.

West. 14 hrs. 51 min. 20-7 sec.

2 ) 25 hrs. 33 min. 51-2 sec.

Meridian transit by chronometer, . 12 hrs. 46 min. 55-6 sec.
R.A. of star, 12 hrs. 42 min. 41 sec.

Chronometer correction, . . . 4 min. 14-6 sec.

As the chronometer is too fast, the correction is to be subtracted from the
chronometer reading.

If, as is more usual, the chronometer keeps local mean
time, the sidereal time at the meridian transit of the star
must be reduced to local mean time in order to compare
with the chronometer time. This cannot be done without
a knowledge of the longitude.

EXAMPLE. — At a place in longitude 8 hrs. 35 min. 27 sec. East, on the
evening of September 1st, 1914, the star a Pavonis is observed East of the
meridian at 7 hrs. 9 min. 20-5 sec., with a watch keeping local mean time.
It is again observed at the same altitude to the West of the meridian at
9 min. 30-2 sec. after midnight. Find the error of the watch, having given
G.S.T. at G.M.N., September 1st, 1914, 10 hrs. 39 min. 13-38 sec.
R.A. of a Pavonis, .... 20 hrs. 18 min. 57-4 sec.

Ans. 8-1 seconds slow.

204 ASTRONOMY FOR SURVEYORS.

It is desirable, in order to make the determination as
precise as possible, that a series of observations should
be made upon the star on each side of the meridian,
instead of one observation only. A few times should be
taken when the star is on the East of the meridian at
altitudes differing by 20 or 30 minutes of arc. A corre-
sponding series of times should then be taken when the
star is on the West of the meridian at the same altitudes.
Since all that we want to ensure is that the altitude is
the same at corresponding observations East and West
of the meridian, there is no particular object in reversing
the face of the instrument. The whole set of observations
may be taken with the one face.

The Error due to a Slight Inequality in the Altitudes of two
Corresponding Observations. — If in Fig. 45 Z S± = zenith dis-
tance of the first observation = z,

Z P = co-latitude = c
P Sj = polar distance = p
h = hour angle Z P Sj_
Z = angle Sx Z P = azimuth of star
cos z = cos c cos p -f- sin c sin p cos ht . ( 1)

Suppose now that at the second observation the zenith
distance, instead of being z, is z + y, being in error by a
small amount y. Then the hour angle Z P S2 will be in
error by a corresponding amount x, so that instead of being
h, it will be h + x. Then, from the spherical triangle Z P S2,
cos (z+ y) = cos c cos p+ sin c sin p . cos (h+ x). (2)

Subtracting (2) from (1), treating x and y as small
quantities, we get

y . sin z = x sin c sin p sin h .
sin z sin p

But

sin h sin Z '

*=-- -V-
sin c sin Z

DETERMINATION OF TIME BY OBSERVATION. 205

We see thus that the error x in the hour angle, corre-
sponding to an error y in the second altitude, will be least
when Z = 90°, and will be greater the smaller the value of Z,
We draw, therefore, the practical conclusion that the ob-
servations are best made on stars near the prime vertical.

If the declination of a star is slightly less than the
latitude, it will cross the prime vertical near the zenith
and the interval between the times of transit will be
small. This, therefore, is a convenient observation to
make, and the conditions are favourable to accuracy.

The Determination of Time by Equal Altitudes of the Sun.
—The above method is an extremely simple one as
applied to the stars, because the -declination of a star
remains constant during the period over which the
observations extend. But in the case of the sun the
declination changes so rapidly that it cannot be considered
as constant, and the theory becomes complicated by
the fact that allowance must be made for the alteration
of declination in the interval between the observations.
Referring again to Fig. 45, if p denotes the polar distance
of the sun when it is on the meridian, then at the first
sight, when the sun is at Sl5 the polar distance will be
p±y, and at the second sight, when the sun is at S2, the
polar distance will be p=f y. The -f or — sign is to be
taken in the first of these expressions according as the sun
is approaching or leaving the elevated pole.

If p were constant, we should have

cos z = cos p cos c + sin p sin c cos h.

But if at the first observation, S1? the polar distance
is p+ y, the hour angle will be h + x, and we have

cos z = cos (p + y) cos c -f sin (p + y) sin c cos (h -f x).

Subtracting these two equations, and treating x and
y as small quantities, we get

0 = y sin p cos c—y cos p sin c cos h + x sin h sin c sin p.
— x = y (cot c cosec h — cot p cot h).

206 ASTRONOMY FOR SURVEYORS.

Under these conditions the first observation will be made
when the sun is at an hour angle h -fa before apparent
noon, where x is given by the preceding expression, and it
may be positive or negative according as cot c cosec h
is < or > cot p cot h.

Similarly the second observation will be made with the
sun at an hour angle h — x after apparent noon, and it may
be shown in the same way as before that the value of x
is given in this case also by the same mathematical
expression.

The mean of these two observed times will therefore
be when the sun is at an hour angle x before apparent
noon.

When the sun is leaving the elevated pole, instead of
approaching it, the mean of the two observed times will be
when the sun is at an hour angle x after apparent noon.

Thus, the true time of transit — i.e., the time of apparent
noon — is given by

Mean of observed times ± yV y (cot c cosec h — cot p
cot h).

y is the alteration in the sun's declination in half the
time interval between the two observations.

h is half the time interval between the two observations
reduced to angular measure.

The + sign is to be taken if the sun is leaving the
elevated pole, and the -- sign when it is approaching
the elevated pole.

Just as with star observations, it is necessary, in order
to obtain the best results, that a series, say four or six,
of observations should be taken to the sun in the forenoon
and a corresponding set in the afternoon, the sights in
each case being taken alternately to the upper and lower
limbs.

EXAMPLE. — At Adelaide, longitude 9 hrs. 14 min. 20 sec. E., latitude
34° 55' 38" S., on July 21st, 1914, equal altitude observations of the sun

DETEKMINATION OF TIME BY OBSERVATION. 207

were taken in the forenoon and afternoon. The means of the noted times
were 9 hrs. 35 min. 03 sec. a.m. and 2 hrs. 37 min. 15 sec, p.m. by a watch
keeping mean time.

12 hrs. 00 min-. 00 sec.
subtract 9 hrs. 35 min. 03 sec.

2 hrs. 24 min. 57 sec.
add 2 hrs. 37 min. 15 sec.

2 ) 5 hrs. 02 min. 12 sec. = time between observations.

2 hrs. 31 min. 06 sec. .'. h=3T 46' 30".
subtract from 2 hrs. 37 min. 15 sec.

0 hr. 6 min. 09 sec. = time by watch at apparent
noon.

c = 55° 04' 22"

Declination at G.A.N., July 21st, . . 20° 36' 02-5"
Correction for longitude, .... 2' 41 -5"

Declination at L.A.N., . . . .20° 38' 44'
.'. p, 110° 38' 44'

cot c . cosec h . . . = 1-140

cot p cot h . . . . = — -486

cot c cosec h— cot p cot h . . = 1-626
Change in declination in 2 hrs. 31 min. 06 sec. = 71-69",
and sun is approaching elevated pole,

1-626 x 71-69

.-. time of apparent noon = 6 09 - — =-= — — seconds

= 6' 09"- 7-6" = 6' 01 -4".

But, from the Nautical Almanac, the equation of time to be added to
apparent time at L.A.N. is 6' 08-3", which is, therefore, the true time of
apparent noon.

Thus the watch is 7 seconds slow.

Fourth Method — -Almucantar Method for Time Observations.

—In 1884 Mr. S. C. Chandler, at the Harvard College
Observatory, U.S.A., devised a form of instrument in
which the telescope was fixed at a constant angle with
the vertical, so that the line of sight traced out a hori-
zontal circle on the celestial sphere, and observations for
the determination of latitude and other purposes were
made by noting the times of transit of stars across the
fixed horizontal circle. The instrument was named an

208

ASTRONOMY FOR SURVEYORS.

" almucantar," and it proved to be capable of very-
remarkable work. The same principle may be readily
applied with an ordinary theodolite, and experience has
shown that extremely accurate determinations of time are
possible in this way.*

Any horizontal circle may be used for the observations,
but the most convenient is the one that passes through
the pole of the observer. This has been named the " co-
latitude circle/' its zenith distance being everywhere equal
to the co-latitude. The formulse for reduction then become
very simple. The method consists in observing the
times of transit of a series of East and West stars, some-
where near the prime vertical, across the horizontal

Fig. 45a.

wire of a telescope that is set to an altitude equal to that
of the pole. Allowance must be made for refraction, and,
therefore, the telescope is actually set so that its altitude
as read off on the vertical circle is equal to the latitude
of the place plus refraction.

In Fig. 45a, Z denotes the zenith, P the celestial pole,
A and B the North and South points, P S Q the co-latitude
circle. Let S denote the position of a star, somewhere
near the prime vertical, as it crosses the co-latitude
circle.

* See paper by W. E. Cooke, " On a New and Accurate Method of deter-
mining Time, Latitude, and Azimuth with a Theodolite " — Monthly Noticee,.
Royal Astronomical Society, January, 1903.

DETERMINATION OF TIME BY OBSERVATION. 209

Let Z P = c = co-latitude.

P S = p= star's polar distance, measured, of course,
along the great circle arc P N S and not along the small
circle P S Q.

Angle S P Z = h = hour angle of star.

Angle S Z P = Z = azimuth of star measured from
elevated pole.

Then, since Z S = c, Z S P is an isosceles triangle, and,
if Z N be drawn perpendicular to the great circle arc
joining S and P, it will divide S Z P into two equal right-
angled triangles.

From the triangle Z N P

cos N P Z = tan P N cot Z P

P P

cos h — tan - . cot c = tan . tan I . (1)

'— 2i

if I is the latitude of the place.

To determine the azimuth at which a star will cross
the co-latitude circle, from the same triangle

cos Z P = cot N Z P cot N P Z.

cos c=cot h . cot — ,

or cot f — sin I . tan h. . . (2)

Formula ( 1 ) enables the time of transit to be com-
puted, and formula (2) gives the azimuth if required.

If an observation on one star in the East is balanced
by a corresponding observation on a star in the West
of somewhere about the same declination, then the mean
of the two time observations will give a correct result
even if the co-latitude circle is considerably out. If, for
instance, the co-latitude circle is set out too low, the
observed time of transit in the East will be too soon, but
that in the West will be too late, and if there is not much

210 ASTRONOMY FOB SURVEYORS.

difference in the declinations of the stars the time of
transit will be just as much too soon in the one case
as it is too late in the other. Thus by averaging the two
results any small error in the setting out of the co-latitude
circle is practically eliminated, and it is not necessary,
therefore, in order to apply the method that the latitude
of the place should be known with precision. An approxi-
mate latitude will suffice.

For precisely the same reasons as have been investi-
gated when dealing with extra-meridian observations for
time, slight errors in latitude, declination, and altitude
will have least effect upon the result when the stars
observed are near the prime vertical. The stars should
be selected from a zone of about 20° on each side of the
prime vertical.

EXAMPLE.— On May 3rd, 1903, in Lai. 31° 56' 45" S., the transit of /?
Orionis was observed in the West across the co-latitude circle at 8 hrs. 55 min .
1 -5 sec. by a watch keeping sidereal time. The transit oj a Virginia icas similarly
observed in the East at 9 hrs. 20 min. 23-4 sec. Determine the error of the
watch.

B Orionis. a Virginis.

Declination, . . 8° 19' 2-7" S. 10° 39' 30-1" S.

p, . . ..'•".. 81°40'57-3" 79°20'29-9"

\ ]>, . - . . 40° 50' 28-6" 39° 40' 15"

log tan |, . . 9-9367323 9-9187412

log tan/, -. . 9*7948752 9-7948752

log cos h, . . . 9-7316075 9-7136164

/*, . . . . 57° 22' 58" 58° 51 '31"

I) in time, . . 3 hrs. 49 min. 32 sec. 3 hrs. 55 min. 26 sec.

II. A. of star, . . 5 hrs. 09 min. 52-6 sec. 13 hrs. 20 min. 07-5 sec.

Computed time, . 8 hrs. 59 min. 24-6 sec. 9 hrs. 24 min. 41 -5 sec.

Observed time, . 8 hrs. 55 min. 01-5 sec. 9 hrs. 20 min. 23-4 sec.

Error of watch (slow), 4 min. 23-1 sec. 4 min. 18-1 sec.

Mean determination of watch error. — 4 min. 20-6 sec. slow.

Adjustment of Telescope during Observation. — It is the
most essential thing for accurate work, in observations

DETERMINATION OF TIME BY OBSERVATION. 211

of this kind, that the telescope should throughout make
exactly the same angle with the horizontal. It is not
of such importance that the -altitude should be exactly
equal to the latitude, ; a,sr it is that the altitude should
remain the same throughout the observations. Now, no
matter how carefully a transit theodolite is adjusted,
the bubble attached to the vertical circle will not remain
precisely in the centre of its run as the telescope is turned
from star to star. It is, therefore, essential to accurate
work that this bubble should be adjusted to the centre
of its run just before the star crosses the horizontal wire
in each case. This must be done, of course, by the ad-
justing screw on every transit theodolite that moves
both telescope and vertical circle together without affecting
the altitude reading. After .the reading on the vertical
circle has been set for the first star so that the altitude is
equal to the latitude plus refraction, the altitude screw
which would alter this reading must on no accdunt be
touched. But at each observation the horizontal line
of the vertical circle must be adjusted without altering
the reading of the vernier.

To get the most accurate results observations must
be made upon a number of stars, at least six in the East
and six in the West, and the mean of all the determinations
is taken. The East and West stars should be selected so
that the angles in azimuth that one set make to the
East are as nearly as possible equal to the angles that
the other set make to the West.

Sun Dials.

Whilst the sun dial does not provide the surveyor with
a means of determining local time with anything like
the precision obtainable by the methods that have been
described, it enables the time to be fixed quite sufficiently
near for the regulation of watches and clocks for ordinary

212 ASTRONOMY FOR SURVEYORS.

purposes, and the instrument may be read just as easily as
a clock. It is especially useful in the remote parts of
sparsely populated countries where no other means of
checking the clock times are available.

When a sun dial is illuminated by the direct light of the
sun the shadow of a straight line or sharp straight edge
is thrown upon a plane containing a graduated circle so
marked that the apparent solar time is indicated by the
reading at the place where the shadow intersects the
circle. The plane containing the graduated circle may
be either horizontal, vertical, or inclined. The straight
edge, the shadow of which is thrown upon the circle, is
always set up so as to be parallel to the earth's axis. It
is called the stile, or gnomon of the dial. When the gradu-
ated circle or " plane of the dial " is horizontal we have
what is known as a horizontal dial, and as this is the
most common form we will consider it first.

The Horizontal Dial. — In Fig. 46, let M B L A represent
the plane of the dial, which we may suppose to be ex-
tended indefinitely so that M B L A is the circle in which
it intersects the celestial sphere. C P is the direction
of the gnomon, which again we may suppose to be produced
to intersect the celestial sphere in the celestial pole P.
B P A is the plane of the meridian.

If now S denotes the position of the sun, the line of
intersection of the shadow of the gnomon C P with the
plane of the dial will be the line of intersection of the
plane containing C P and S with the plane M B L A.
MPL represents in the figure the plane passing through
S and C P, and M C L is the line of intersection of this
plane with the plane of the dial, or C L is the direction
of the shadow of the gnomon.

Neglecting the slight alteration in the declination of
the sun during the hours of daylight, S will describe
a circle uniformly on the celestial sphere about P as
centre. The angle S P B is the hour angle of the sun,

DETERMINATION OF TIME BY OBSERVATION. 213

decreasing or increasing uniformly with the time according
as the observation is made in the morning or in the after-
noon.

Then in the right-angled triangle L P A

A P = I = latitude of place.
Angle A P L = h = hour angle of sun.

A L = x = required division along the dial
corresponding to hour angle h.
sin 1= cot h tan x, or tan x= sin / tan h.

Thus, to graduate the dial for the hourly intervals
before and after noon, we must put h= 15°, 30°, 45°,

etc., in succession and compute the corresponding values
of x, knowing, of course, the value of /.

Thus, if the latitude of the place is 30°, the first hourly
division on each side of noon will be marked out at an
angle with C A given by

log tan x = log sin 30° -f log tan 15°,

from which x= 7° 38'.

The next hourly division, indicating either 10 a.m. or
2 p.m. will make an angle with C A given by

log tan x = log sin 30° + log tan 30°,
from which x— 16° 6', and so on.

The reading of the shadow of the gnomon gives the

214

ASTRONOMY FOR SURVEYORS.

local apparent time which must be corrected by the equa-
tion of time, as given by the Nautical Almanac, in order
to obtain the mean time. A table of corrections may
easily be drawn out for different times of the year.

The Prime Vertical Dial. — In this case the plane of the
dial lies in the prime vertical. In Fig. 47 let A L B M
be the plane of the dial, which we will again suppose is
continued on indefinitely, so as to cut the celestial sphere.
C P, the direction of the stile or gnomon, is again parallel

to the earth's axis, but this
time P will be the celestial pole
below the visible horizon. APB
is the plane of the meridian.

Then if, as in the previous
case, S denotes the position of
the sun on the celestial sphere,
the apparent movement of S is
to describe a circle on the
celestial sphere with P as
centre, and the hour angle of
S is the angle SPA.

The shadow of P C thrown
by S upon the plane of the
dial will be C M, the line of
intersection of the plane passing
through S and P C with the plane of the dial.
In the right-angled spherical triangle P B M
P B = 90° - / = co-latitude.
Angle B P M = h = hour angle of sun.

B M = x = required division along the dial correspond-
ing to the hour angle h.

cos I = cot h tan x

or tan x = cos I tan ^,

and by this formula the dial may be graduated in a similar
manner to the horizontal dial.

Fig. 47.

DETERMINATION OF TIME BY OBSERVATION. 215

Oblique Eials. — If the plane of the dial is inclined to
the horizontal the dial is said to be " oblique/' There
is one case that is particularly simple, and has given
rise to some of the simplest sun dial constructions. This
is the case in which the plane of the dial is tilted so as
to be perpendicular to the stile, so that it coincides with
the plane of the celestial equator. With this arrangement
the shadow of the stile on the dial moves round uniformly
with the revolution of the sun and the hour divisions
on the dial are consequently uniformly spaced.

Fig. 48.

Time of Rising or Setting of a Celestial Body.

This is not of much value for the determination of
time, because of the uncertainty of refraction on the
horizon. In Fig. 48, if A S B be the plane of the horizon,
Z the zenith, P the celestial pole, and S the body, which
is exactly on the celestial horizon, then the spherical
triangle P S A is right-angled at A, and

cos SPA=cot SP tan PA.

cos (hour angle S P Z) = — tan dec. tan lat.

216 ASTRONOMY FOR SURVEYORS.

From this the hour angle of the body at rising or setting
may be computed, and this will determine the apparent
solar time in the case of the sun or the sidereal time if
a star is observed.

We have here neglected the effect of refraction, which,
amounting as it does to about 36' on the horizon, will
cause stars to be just visible when they are really 36'
below the horizon.

To find the azimuth of the body, we have

cos S P = cos S A cos P A,
sin dec.

or cos S A =

cos lat.'

EXAMPLES.

1. At a place in lat. 35° S., the bearing of a wall is 1 10°. Find the apparent
time at the equinox when it casts no shadow.

Ans. 3 hrs. 50 min. 24-5 sec. p.m.

2. Find the true bearing and apparent time of sunrise in lat. 32° S. when
the sun's declination is 20° S. (Take the sun's centre and neglect refraction
and parallax.)

Ans. Bearing, 113° 47' 05".

Time, 5 hrs. 07 min. 25 sec.

3. Rigel was observed East of the meridian on the horizontal wire of a
theodolite at 7 hrs. 05 min. 20 sec. p.m. by a watch which is supposed to
keep West Australian standard time (120th meridian). It was also observed
at the same altitude West to cross the horizontal wire at 1 hr. 25 min. 30 sec.
a.m. Neglecting the rate of the watch, find its error.

Date of first observation, . . . January 5th, 1908.

Longitude of locality, .... 1 15° 50' 26" E.

Sidereal time at G.M.N., January 5th, . 18 hrs. 54 min. 45-83 sec.

Sidereal time at G.M.N., January 6th, . 18 hrs. 58 min. 42-39 sec.

R.A. of Rigel, 5 hrs. 10 min. 07-29 sec.

Ans. 16 min. 9-8 sec. slow.

4. On July 16th, 1910, in latitude 33° 15' 13" S. and longitude 10 hrs.
04 min. 50 sec. E., the observed altitude of the sun's centre was 31° 54' 45"
bearing 10° 35' 15" magnetic, the referring mark bearing 86° 54' 15"
magnetic, time by watch being 10 hrs. 48 min.

DETERMINATION OF TIME BY OBSERVATION. 217

The sun's declination at noon on July 15th at Greenwich was 21° 38' 18"
N., and the mean hourly difference 23-05" decreasing.

The equation of time to be added to apparent time is 5 min. 46-18 sec.,
and the hourly increase 0-25 sec.

Find the true bearing of the referring mark, the magnetic variation,
and the error of the watch.

Ans. Bearing, 98° 34' 46".

Variation, 11° 40' 31" E.
Watch error, 3' 04-3" fast.

5. At a place 40° 51' 20" S., 140° 20' 30" E., at 9 hrs. 10 min. 20 sec.
a.m. by a watch on 2nd September, 1910, the sun's preceding limb was
found by compass bearing to be 58° 14' 20", and the observed altitude of
the upper limb 27° 11' 15".

Declination at G.M.N., September 1st, 8° 31' 00-7" N. ; hourly variation,
54-24".

Declination at G.M.N., September 2nd, 8° 09' 14-4" N. ; hourly variation,
54-58".

Sun's semi-diameter, G.M.N., September 1st, 15' 52-61".
September 2nd, 15' 52-84".

Equation of time (to be added to apparent time), G.A.N., September 1st,
9-04 sec.

Equation of time (to be subtracted from apparent time), G.A.N., Sep-
tember 2nd, 9-66 sec.

What was the declination of the compass and the correct mean time of
observation ?

Ans. Declination, 9° 21' 15" West.
Mean time, 9 hrs. 07 min.
57 sec.

6. At a place in latitude 32° S. a vertical rod 6 feet high casts a shadow
15 feet long in a direction bearing 75° 12'. What is the apparent time and
the approximate time of year ?

Ans. 5 hrs. 5£ min. p.m.
December.

7. If the time be found by a single altitude, show that a small error in
the latitude will have no effect on the time when the body is in the prime
vertical.

8. At 5 p.m. by watch on September 8th at a place in latitude 31° 57'
08-4" S., longitude 7 hrs. 43 min. E., the observed altitude of the sun's
centre (corrected for instrumental errors) was 29° 58' 25-2". Sun's declina-
tion at G.A.N., September 8th = 5° 45' 55-9" N., variation in one hour
56-40".

218

ASTRONOMY FOR SURVEYORS.

Equation of time to be subtracted from apparent time = 2 min. 18 sec.
Find the sun's true bearing and the error of the watch on West Australian
standard time (120th meridian).

AIM. Bearing, 299° 49' 06-32".

9. On January 3rd, 1914, at a place latitude 30° 15' S., longitude 148° E.,
the following sun observation was taken : —

Observed Altitude.

Alidade Approximate Local Mean A,,WIA fr-,m T? \r
Bubble. Time by Watch. roui B.M.

|O 38° 07' 15"

E. 0.

37 8 hrs. 6 min. a.m. 112° 14' 40"

O, 39° 18' 37"

2 8 8 hrs. 10 min. a.m. 114° 51' 20"

Magnetic bearing of R.M , 200° 10' 20".

Bubble divisions on Alidade = 20".

Required : Magnetic Variation and Error of Watch .

Data from Nautical Almanac : —

Sun's Declination. Hourly Variation.

Jan. 3rd, G.M.N., 22° 53' 02-4" S., . . . 14-08"
Jan. 4th, G.M.N., 22° 47' 11-0" S., . . . 15-21"
Equation of time (to be added to apparent time).

Jan. 3rd, G.M.N., 4 min. 23-81 sec., . . 1-162"

Jan. 4th, G.M.N., 4 min. 51-51 sec., . . 1-145"

Ans. Magnetic variation = 9° 44

11" E.

Error of watch = 6 min.
42 sec. slow.

219

CHAPTER XI.

DETERMINATION OF LONGITUDE.

THE difference of longitude between any two places on
the earth's surface, as we have already seen, is measured
by the difference between either their local sidereal
times or their local mean times at the same instant.
The problem, then, of the determination of the difference
in longitude between A and B amounts to that of the
determination of the difference in the local times at A and
B. By the methods we have considered in the last
chapter we may by astronomical observation determine
the local time at A at some instant, and a means must
be found of determining what is the local time at B
at the same instant, if we are to ascertain the difference
of longitude.

The problem presented is usually that of the deter-
mination of the difference of longitude between two
places rather than the fixing of the absolute longitude
of a place as measured from the now universal standard
meridian, that of Greenwich. Usually we seek to find
the difference in longitude between a point on a survey
and some fixed observatory in the country or some other
point on the survey, the longitude of which has been
previously determined.

In all cases the local time at some instant must be
determined at the place whose longitude is required
by one of the astronomical methods of the last chapter.
The corresponding local time at the reference station

220 ASTRONOMY FOR SURVEYORS.

is then in modern practice usually found by one of three
ways :—

(a) By portable chronometers.

(b) By electric telegraph or wireless telegraphy.

(a) By Portable Chronometers. — Since the time when
chronometers that will retain a fairly uniform rate have
been generally available, this has been the general method
for the determination of longitude at sea. Every ship
carries a chronometer, which keeps either Greenwich
time or the local time at some known port, and from
an astronomical observation the Captain is thus able to
ascertain the difference between his local time and that
of the chronometer. The method is very simple and con-
venient, but wireless telegraphy, which is capable of much
greater precision, may perhaps largely supersede it in
the near future. To obtain accurate results it is essential
that the chronometer should keep a constant rate, and
the conditions on board a ship are much more favourable
for this than is usually the case when chronometers are
carried about from place to place on land. So that for
land work the box chronometers used at sea are com-
monly replaced by chronometer watches which are more
easily carried and are found to be more satisfactory.

Suppose now that it is required to determine the
difference in longitude between A and B. The watch
or chronometer must first be regulated at station A.
Its error on the local time at that place must be deter-
mined and its "rate"— i.e., the amount that it gains
or loses in 24 hours — must be found. On the assumption
that the rate remains constant this will enable the local
time at A to be found from a reading of the chronometer
at any time afterwards. If then the chronometer be
transported to B and an astronomical observation be
made there for the determination of local time, it will

DETERMINATION OF LONGITUDE. 221

be possible to find from the chronometer the local time
at A at the same instant.

EXAMPLE. — At A, September 8th, 1914, the chronometer at 8 p.m. was
found to be 2 min. 6-5 sec. fast, and it was gaining at the rate of 2-58 sec.
in 24 chronometer hours.

At B, September 9th, 1914, from an astronomical observation which
gave the local time as 9 hrs. 12 min. 35 sec. p.m., the reading of the chrono-
meter was 9 hrs. 12 min. 30-6 sec.

What is the difference of longitude ?

The interval of time, as measured on the chronometer, between the two
readings is 25 hrs. 10 min. 24-1 sec. = 1-049 days.

Therefore, in this interval the chronometer has gained 1-049 X 2-58 sec.
= 2-7 sec.

Thus, at B the chronometer was fast by 2 min. 9-2 sec., and the local
time at A was 9 hrs. 10 min. 21 -4 sec., corresponding to the local time of
9 hrs. 12 min. 35 sec. at B.

Thus, the time at B is in advance of that at A by 2 min. 13-6 sec., or
B is to the East of A by 0° 33' 24".

The accuracy of the method is affected by the fact
that the rates of chronometers are not perfectly constant,
and particularly by the fact that the rate whilst being
carried is not the same as when at rest. The best way
to minimise the error is to use several chronometers,
from each of which a longitude determination is obtained,
and the average of the results is taken. If possible,
after the observations have been made at B, the chrono-
meters should be carried back again to A and another
comparison made with the local time there.

This method is now never used by surveyors except
where telegraphic communication is not available.

(b) By Electric Telegraph or Wireless Telegraphy. — If two-
places are connected by electric telegraph the difference
of longitude may be obtained with great accuracy.

Suppose that A and B are two stations so connected,.
A being to the east of B, so that the local time at A i&
in advance of that at B.

Then if an operator at A taps a telegraphic key that

222 ASTRONOMY FOR SURVEYORS.

produces a corresponding tap in a telegraphic key at B,
the two taps will be very nearly simultaneous, but not
quite. A certain slight interval of time, a fraction of a
second, will be required to transit the electric current
from A to B and to produce the motion of the recording
instruments. But whether the signal be transmitted
from A to B or in the reverse direction from B to A, the
time taken in transmission will be the same.

If now the operators at A and B note the exact instant
of each tap on chronometers keeping local time, either
mean solar or sidereal, the difference in the times would
at once give the difference in longitude if the taps were
absolutely simultaneous.

But, actually, when the message is sent from A to B,
owing to the time taken in transmission, the tap at B
will be a little later than it should, and the result obtained
for the difference in longitude will be correspondingly
too small.

And similarly when the message is sent from B to A,
the tap at A will be made later than should be the case
if the transmission were instantaneous, and A being to
the east of B, the difference of time will now appear too
great.

Thus by averaging the results of sending messages in
opposite directions a correct value is obtained for the
difference in longitude, and the error due to the time of
transmission is completely eliminated.

With signals sent by wireless telegraphy the velocity
of the electric wave is so great that practically there is
no measurable difference in the results obtained, whether
the signals are sent from A to B or from B to A.

For the most refined determinations the signals as
received are automatically recorded on a chronograph,
but very good work can be done by noting the times
of signals with a chronometer if proper methods are
adopted .

DETERMINATION OF LONGITUDE. 223

Recording and Receiving Signals. — A set of signals usually
consists of a series of taps made at intervals of 10
seconds by a sidereal chronometer, the set extending
over from 3 to 5 minutes. Each set is ushered in by a
warning rattle of the key. The exact time of each tap
is recorded at the receiving station by an observer who
is counting out the ticks, which represent half seconds,
on a chronometer keeping mean time. If the tap occurs
between 1-5 and 2-0 seconds, the observer judges whether
the time is 1-6, 1-7, 1-8, or 1-9.

It is a very important aid to accuracy that the 10
second signals should be sent by means of a sidereal
chronometer and recorded by a mean time chronometer.
If the chronometer at the sending and receiving ends
kept the same kind of time, the taps would always occur
at the same decimal of a second, and the recorder, after
the first two or three taps, would probably become pre-
judiced in favour of some particular value of the decimal
which he would retain throughout the set. But if one
chronometer keeps sidereal and the other mean time,
the tick of the sidereal chronometer Avill coincide with
that of the mean time chronometer every three minutes,
and in the interval between the coincidences the deci-
mals of a second recorded at the receiving station will
range from -1 to -9, so that the judgment of the recorder
is not likely to be prejudiced in the same way as it would
be if both instruments kept the same kind of time.

Comparison of Chronometers. — If two chronometers keep-
ing the same kind of time, both beating half seconds,
are to be compared, it will generally happen that the
ticks of the one do not exactly coincide with the ticks
of the other, but differ by some fraction of a half second
that must be estimated by ear. It is difficult and re-
quires considerable practice to make this estimate nearer
than the fifth of a second. But it is possible to compare
a sidereal .and a mean time chronometer with much

224 ASTRONOMY FOR SURVEYORS.

greater accuracy, because at intervals of about three
minutes the ticks of the two exactly coincide, and, if
the comparison be made at the moment of coincidence,
there is no difference of a fraction of a beat for the ear
to estimate. Thus the difference in the readings of the
two chronometers at this particular instant may be
obtained exactly. The only error will be that which
arises from judging the beats to be in coincidence when
they are really separated by a small fraction. But it is
found that a difference between the beats as small as
0-02 second is sufficient to enable the practised ear to
detect the departure from exact synchronism and con-
sequently the comparison may be made with an error not
exceeding this quantity.

The error of the sidereal chronometer is first obtained
by astronomical observation, in the manner described
in the previous chapter. Then to determine the error
of the mean time chronometer a comparison is made
at one of the moments when the beats coincide. List-
ening to the beats of the two chronometers the observer
judges when a coincidence is about to occur. He then
begins to count^ the beats of one chronometer while he
watches the face of the other. When he no longer per-
ceives any difference in the beats, he notes the corre-
sponding half seconds of the two instruments. The
observed instant on the sidereal chronometer is then
reduced to mean time, after allowing for the error of the
chronometer, and the difference between the result and the
recorded instant on the mean time chronometer gives its
error.

Personal Equation. — It is found that different men,
when performing such operations as sending or record-
ing signals, will differ appreciably in their work. One
man, when pressing down a telegraphic key at the instant
the chronometer ticks, will consistently do so a little
too late. Another will invariably press the key a small

DETERMINATION OF LONGITUDE. 225

fraction of a second too soon. Similarly when recording
the time signals one observer will consistently make a
larger error than the other. II is found that the more
practised and experienced the observers are, the more
regular and consistent are the errors made in this way,
and that this personal error or " personal equation/'
as it is commonly called, remains fairly constant for
long periods of time. Consequently its effects may be
largely eliminated, in the average of a considerable
number of observations, if the personal equations of the
observers be determined both before and after the obser-
vations are made.

In this case the relative personal equation is required
between two observers. It may be most simply obtained
by the observers setting up their instruments near to
one another at the same station. They then send sets
of signals to one another, just as they would do in
ordinary field work, in order to determine their difference
of longitude. This should be done under conditions as
nearly as possible the same as those obtaining at the
actual work in the field. The result obtained, which
should of course be zero, is the relative personal equation
that must be applied in the reduction of the field obser-
vations. It is advisable to observe the personal equation
in this way for two or three evenings shortly preceding
and following the field trip.

When a large number of observations is being made
probably the best way of eliminating the error due to
personal equation is to exchange the observers at the enda
of the telegraph line when half the total number of
signals have been transmitted. When A sends and B
receives, the time recorded at the receiving station should
exactly coincide with the time of sending. Usually it
does not, owing to the existence of this personal equation,
and the time actually recorded by B may be either before
or after the chronometer tick that A is transmitting.

226 ASTRONOMY FOR SURVEYORS.

If the time recorded is always after the chronometer
tick, the error will be fairly consistent so long as A is
sending and B receiving. If B is at a station to the
east of A, the effect of this error will be to make the
difference of longitude greater than it really is, but if
B is at a station to the west of A the same error will
make the difference of longitude appear less than it should
be. Thus if the observers change places when half the
observations are over, personal equation is eliminated in
the mean of the whole set and there is no necessity to
make a special determination of it.

Programme of Operations. — Observations are made on
several evenings. Professor W. E. Cooke, who was
responsible for the introduction of the almucantar
method of time observation in Western Australiai thus
summarises the operations for any one evening : —

Observations.

(a) Compare sidereal and mean time chronometers.

(b) Take first half of almucantar observations, using
sidereal chronometer.

(d) Complete almucantar observations.

(e) Compare the two chronometers.

Computations.

(/) From the almucantar observations determine the
error of the sidereal chronometer at some definite sidereal
hour, also its rate.

(g) Apply the rate so as to obtain the error at time (a) ;
reduce sidereal time (a) to mean, and hence determine
error of mean time chronometer at time (a).

(h) Do the same for time (e).

(i) From (g) and (h) determine the errors of each
chronometer at time (c).

DETERMINATION OF LONGITUDE. 227

(j) Apply these errors to the average of the signals,
also apply the correction for personal equation. Sub-
tract the results from the similar results at the other
station, and thus the difference of longitude will be
obtained.

When a determination of difference of longitude is
made telegraphically between fixed observatories, the
precision of the method is increased by sending the
signals from a clock, the pendulum of which automatically
completes an electric circuit when at the bottom of its
stroke. The record at the other station is then taken on
a chronograph, from which the instant can be read off to
the hundredth part of a second. Such equipment is,
however, not usually available for field work.

(c) By Flash-Light Signals. — When two stations are visible
one from the other, flash light signals may be sent from
one at ten second intervals as determined by the tick of
a sidereal chronometer and recorded at the other by
means of a chronometer keeping mean time, just as with
electric telegraph signals. Or the signals may be sent
from an intermediate station that is visible from both.
The observers at each station must of course have
obtained their local time by proper observation, and the
difference between their local times at the instant of the
signal gives at once the difference of longitude. The
signal may be made by the flash of a heliotrope by day
or the eclipse of a bright light at night.

The following examples gives the results of obser-
vations made in this way in Western Australia to de-
termine the difference of longitude between the Perth
Observatory and Mount Maxwell, about 17 miles away
to the east. The signals were made by means of an
acetylene lamp placed in a box, the light shining through
a hole over which a photographic snap-shutter was fixed.
The shutter was released at the proper second and the
time of the flash noted as it was seen through a theodolite

228

ASTRONOMY FOR SURVEYORS.

at the other station. The example is taken from the
Western Australian Handbook for Surveyors : —

DIFFERENCE OF LONGITUDE.

1909.

Mount Maxwell j Observatory to
to Observatory. Mount Maxwell.

Mean Result.

Nov. 6th,
Nov. 7th,
Nov. 8th, . : .. .
Nov. 9th,
Nov. 13th,

7-96' I7 8-64'
7-87' l'S-56'
7-82' 1' 8-54'
7-81' I7 8-61'
7-93' I' 8-53'

i' s-so"

I' 8-21"

rs-18"

1'8-21"

rs-23"

Mean, ....
Personal equation,

l'8-25"
+ 0'0-06"

Difference of time,

1'8-31"

Longitude by Lunar Observations. — The methods for the
determination of longitude that have just been described
are those nowadays most usually adopted, but before
the invention of the electric telegraph and the perfection
of chronometers the only methods available over long
distances depended upon observations of the moon. The
moon changes its position among the fixed stars much
more rapidly than any other celestial body, its relative
movement amounting to over 13° in 24 hours, or roughly
it moves over a distance equal to its own diameter in one
hour. Consequently it is possible to use it as a clock,
and, by measuring its position with regard to surrounding
stars, we may determine at any instant, with the aid
of the tables of the moon's motion given in the Nautical
Almanac, the corresponding time at Greenwich. It was
chiefly in order that " the moon's motion might be
systematically observed for the purpose of providing
navigators with accurate tables, which could be used for
the determination of longitude, that the Greenwich
observatory was originally founded. Lunar observa-

DETERMINATION OF LONGITUDE. 229

tions, however, generally entail rather laborious com-
putation, and the results, with the exception of those
obtained by the method of lunar occupations, are not
comparable in accuracy with the determinations made
by the simpler methods previously given. Consequently
such methods are now rarely used on land, and we shall
merely describe the general principles involved.

There are three principal methods of making observa-
tions upon the moon for longitude. They are :—

(a) By Lunar Distances.

(b) By Lunar Culminations.

(a) By Lunar Distances. — The angular distance between
the bright limb of the moon and some bright star in
its vicinity is measured by means of the sextant, and
at the same instant the altitudes of both moon and star
are observed. This is best done by three observers,
one for each measurement, but if there is only one ob-
server, he takes first the altitudes, then the lunar distance,
and then the altitudes once more, noting the time of each
observation. From these he readily deduces the proper
altitudes at the moment when the lunar distance was
measured.

By adding or subtracting to the observed distance the
apparent semi-diameter of the moon, according as the
bright limb of the moon is toward or from the star, the
apparent distance between the star and the moon's centre
is found. The moon's semi-diameter is given on page
3 of each month in the Nautical Almanac, for noon and
midnight of each day. From this apparent distance,
allowing for refraction and parallax, and knowing the
approximate latitude of the place, the observations
enable the distance to be computed as it would be
observed from the centre of the earth, or the true distance
as it is commonly termed. But if we know the true

230 ASTRONOMY FOR SURVEYORS.

distance the corresponding time at Greenwich may be
found from the information given in the Nautical
Almanac. And the local time of the observation is
readily found from the observed altitude of either moon
or star. The longitude is found, of course, as the differ-
ence between the local time and the corresponding
Greenwich time.

In fig. 49 let S and M denote the apparent positions
of the star and the moon's centre respectively, Z being
the Zenith. Parallax and Refraction will affect them
in the vertical planes Z S and Z M. Now refraction
causes a body to appear at a higher altitude than it
really has, whilst a body when viewed from the earth's
centre will have a greater altitude
than when seen from the earth's
surface. Thus to allow for refrac-
tion we have to decrease the
observed altitude, and to allow
for parallax we must increase it.
Now in the case of the moon
parallax is greater than refrac-
tion, the contrary being true for
a star or planet. Thus the " true "
position of S, as observed from the

Fijr 49.

earth's centre, is at Sl5 below S,
and the true position of M is at M1? above M.

In the triangle Z S M, the three sides have been
directly determined by observation, and, therefore, the
angle Z may be computed by the ordinary rules of
spherical trigonometry. Then in the triangle Z Sl Mx,
Z Sx, and Z M1? are known, and also the included angle
Z, consequently the true 'distance Mj Sj may be computed.

The Nautical Almanac used to give a table of true
lunar distances, for every third hour of Greenwich mean
time, from selected suitable bright stars. But these
tables have lately been discontinued as it was decided

DETERMINATION OF LONGITUDE. 231

that they were no longer of sufficient use to warrant their
retention.

The method is not capable of any degree of precision,
about 5 seconds of time representing the accuracy
attainable, and, now that the tables of lunar distances are
no longer published, involves a lot of computation. The
measurements cannot be made by a theodolite, the
sextant being essential, and the method can only be classed
as a rough one under the best circumstances.

(b) By Lunar Culminations.— As the moon moves right
round the earth in a lunar month of about 28 days, its
right ascension must change by 360° in that period, or
at an average of about 13° in 24 hours. Thus in one
hour its right ascension will alter on the average by
something over 30 minutes of arc or two minutes of time.
Now the right ascension of the moon may be most easily
measured by observing the difference in time between
its transit across the meridian and that of some known
star. If the local time at the place of observation is
also known, this determines the right ascension of the
moon at a given instant of local time. But the Nautical
Almanac gives the right ascension of the moon for every
hour of Greenwich time throughout the year, and, by
interpolation between the values in the tables, the
Greenwich time corresponding to the measured right
ascension may be found. Then the difference between
the local time of observation and the corresponding
Greenwich time as thus determined gives the longitude
required. The computations are thus simple, and the
method is the easiest of all the lunar methods for finding
longitude.

The observations are facilitated by the tables of moon-
culminating stars given in the Nautical Almanac on p.
412 and succeeding pages. In these tables for each day
in the year there are tabulated one or two stars, known
as moon-culminating stars, that do not differ much from

232 ASTRONOMY FOR SURVEYORS.

the moon in either right ascension or declination, and
are consequently suitable for meridian transit observa-
tions in comparison with the moon. For if the declina-
tion of the observed star does not differ much from that
of the moon, any error in the setting out of the meridian
will affect the times of both transits to the same extent,
and in the difference between the two times of transit,
which is what is sought, the error will be eliminated.

The times of meridian transit are unaffected by
parallax and refraction which introduce complications
in other lunar methods. A disadvantage is that for a
considerable part of the month transits occur at very
inconvenient times.

The method in any case is not capable of great
accuracy. An error of one second in the measurement
of the time of transit of the moon's limb will cause
an error of about 30 seconds of time in the longitude.
Thus a good observation will only determine the longi-
tude within about 10 seconds of time, and only by the
average of a number of careful observations will it be
possible to determine the longitude by this method
within 5 seconds of time, corresponding to IJ minutes
of arc, or to a distance of over one mile near the equator.

EXAMPLE. — At a place in approximate longitude 9 hrs. 06 min. E. the
times of transit across the meridian of the moon's bright limb and of the star
y Aquarii icere recorded by means of a chronometer keeping local mean time
on the evening of September 30th, 1914.

Observed time of transit of Moon I.*, . 9 hrs. 14 min. 22-8 sec.
„ „ a Aquarii, . 9 hrs. 52 min. 30-2 sec.

Determine the longitude of the place.

Difference in times of transit, . . 38 min. 07-4 sec.

Equivalent interval of sidereal time, . 38 min. 13-66 sec.

R. A. of r Aquarii, . . . . 22 hrs. 26 min. 09-87 sec.

R.A. of Moon I., 21 hrs. 47 min. ott-21 sec.

* The Roman numerals I. and II. are used in the Nautical Almanac
to indicate the moon's preceding and following limbs respectively.

DETERMINATION OF LONGITUDE. 233

Allowing for the approximate longitude, the transit takes place at about
8 minutes after Greenwich noon on September 30th.
From the Nautical Almanac we obtain

Time of Meridian Sidereal Time of

Passage at Semi-diameter

Greenwich. Passing Meridian.

Sept. 30th, . . 9hrs. 32-1 min. (upper) 63-92 seconds
Sept. 29th, . . 21 hrs. 10-1 min. (lower) 65-02

Thus, the sidereal time for the semi-diameter to pass the meridian is
given by

63-92 + - .^ =

33 hrs. 32 mm. — 21 hrs. 10 mm.

.•. R.A. of moon's centre at instant of observation

= 21 hrs. 49 min. — 00-98 sec.
Again, from the Nautical Almanac,

R.A. of moon. at Greenwich, 0 hr. = 21 hrs. 48 min. 44-20 sec.
1 hr. --= 21 hrs. 50 min. 41-41 sec.

Therefore, by interpolation, the Greenwich mean time corresponding to
the R.A. of 21 hrs. 49 min. 00-98 sec. is

0 hr. 08 min. 35-4 sec.
But the observed local time of the observation is

9 hrs. 14 min. 22-8 sec.
Therefore, the longitude is 9 hrs. 05 min. 47-4 sec. East.

(c) By Lunar Occultations. — In the course of its monthly
revolution round the earth the moon covers or " occults "
in turn a number of the fixed stars. As the moon ap-
parently moves from West to East among the stars, the
stars in its track first disappear under the Eastern
limb and afterwards reappear on the other side. The
covering of a star in this way by the moon is known
as an " occupation," the disappearance of the star
behind the Eastern limb of the moon being known as the
" immersion/' and its reappearance as the " emersion/'
The method by lunar occultations consists in observing
the local time of immersion or emersion, or both, at the
occupation of a known star. At such moments the
apparent right ascension of the star is the same as that
of the Eastern or Western limb of the moon, and, after
making proper allowance for refraction, parallax, and semi-
diameter, the true right ascension of the moon may be

234 ASTRONOMY FOR SURVEYORS.

determined at the instant, and hence, from the tables
in the Nautical Almanac, the corresponding Greenwich
time may be found.

The method is capable of much greater accuracy than
any other method by lunar observations. The two
methods previously described, even under the most
favourable conditions, can give but roughly approxi-
mate results. But from several observations of lunar
occultations a longitude may be determined within less
than one second of time. Unfortunately, however, the
prediction of the circumstances of an occultation and
the complete computation of the observations involve
principles that are rather complex for an elementary
work. Partly on this account, and partly because suit-
able observations can only be made at any one place
some three or four times in a month as a rule, the method
is not one used to any extent by surveyors, and no
further elaboration of the method will in consequence be
attempted here.

Relative Accuracy of Different Methods. — Major Close, in
his Text Book of Topographical Surveying, gives the fol-
lowing table showing the terminal error in longitude which
might be expected after a march of 300 miles in a hilly
tropical country.

Method. Probable Error in Longitude.

Triangulation, . . . .100 yards to J mile.

Telegraph, \ -to \ ; mile.

Chronometers, . . .1 mile.

Occultation, . . . . \ mile.

Moon culminations, . . .1 mile.

Lunar distance, . . . .10 miles.

The probable errors "are here stated as distances
measured parallel to the equator, but, as the actual
measurements of longitude are made in time, and as the
distance measured along the earth's surface correspond-
ing to a given difference of time gets less and less as we

DETERMINATION OF LONGITUDE. 235

proceed further from the equator, it follows that the
probable errors in distance would be considerably less than
those chronicled at places remote from the equator.

Where a triangulation can be carried on to directly
connect the two places whose difference of longitude is-
required, the determination may be made with the greatest
precision possible. The telegraphic method comes next
in order of accuracy, and is nowadays the method most
commonly used. In order to get anything like the same
accuracy by the method of lunar occultations, the observa-
tions would have to extend over several months, and the
tabulated values for the right ascension of the moon given
in the Nautical Almanac would have to be corrected
from observations made at some fixed observatory.

236

CHAPTER XII.

THE CONVERGENCE OF MERIDIANS.

THE line of sight of the telescope of a theodolite in ac-
curate adjustment, as the telescope is turned about its
horizontal axis, traces out a vertical plane. This, if we
regard the earth as spherical, we may consider to be
a plane passing through the centre of the earth. There-
fore, the straight line that is set out by a theodolite is in
reality always the arc of a great circle on the earth's
surface. Now, unless it happens to coincide with the
equator or with a meridian of longitude, any great circle
will cut different meridians at different angles. In other
words, its bearing will vary from point to point. Thus as
we proceed along a straight line set out by a theodolite on
the earth's surface, the bearing of the line will not remain
constant but will gradually alter. A line the bearing
of which was everywhere the same would not be a straight
line. A parallel of latitude for instance is such a line,
but if the telescope of a theodolite is set out truly East
and West at any place its direction would not mark out
the parallel of latitude, which is a small circle, but a great
circle that would ultimately intersect the equator.

This alteration in the bearing of a straight line is an
important matter in surveys of any magnitude, as in
latitudes in the neighbourhood of 60° it amounts to con-
siderably over a minute of arc in a line one mile long,
and in higher latitudes the alteration is still greater.

In fig. 50, let N and S denote the North and South
terrestrial poles, E L M Q is the equator, and A and B

THE CONVERGENCE OF MERIDIANS.

237

any two points between which the great circle arc A B
has been set out.

Let N A M S and N B L S be the meridians through A
and B. Then the bearing of the line B A at B is the
angle NBA, and the bearing of the same line at A is
180° -NAB.

The difference between the bearings of the line A B
at the points A and B is known as the convergence of the
meridians between A and B.

If A B is plotted as a straight line on a plane, then the

meridians through A and B will not be drawn as parallel
lines, but as lines making an angle with one another equal
to the convergence.

Denote the convergence by c.
Then c = 180° - N A B - N B A.

Let /= latitude of A and V = latitude of B.
NA=90°-/, NB=90°— /'.

Denote the difference of longitude between A andp*
by m, so that m = angle B N A.

238 ASTRONOMY FOR SURVEYORS.

Then in the spherical triangle NBA, having given
two sides and the included angle,
tan \ rNBA+NAB)

_ cos \ (NB-NA) cot \rn

cos I (NB+NAJT
.-. cot } (180°- NB A-N AB)

cos \ (I — I') cot I m

or, inverting

tan i

COS i

cos \

i (i8o0 .- I - r)'
UtlJQoaU-f,

— . ..
sin I

sin i

L r

\d+l')

L<L±£)t l4,

I C

1 /I 7/X tai1 2 W.

cos

In any ordinary survey, the length of the line A B
will be very small compared to the earth's radius, and
the angles c and m will be so small that tan \ c and
tan | m may be replaced by J c and | m respectively
without appreciable error.
.-. c (in circular measure)

sin \(l+ V)

- m (in circular measure),
cos | (I — I')

and c (in seconds of arc)

sin i (I + /')

m (in seconds of arc).
cos | (I — I')

Again, unless the line A B is a very long one,
cos J (I —I'} differs from unity by but a very small
quantity, so that for ordinary purposes

Convergence in seconds = sin mid. lat. x diff. of long.
in seconds.

Another convenient form of the result expresses the
convergence in terms of the " departure " between A

THE CONVERGENCE OF MERIDIANS. 239

and B ; that is to say, their distance apart measured in
an East and West direction.

The parallel of middle latitude is a circle of radius
r cos | (I + I'), where r is the radius of the earth in miles,
and, therefore, if d denotes the departure in miles,

r cos I (I + I'
.-. convergence in seconds

= sin J (I + I')

= the circular measure of m.

r cos \ (I + V) sin 1"
d tan | (I + l'\
r sin V

Taking r as 3,958 miles we obtain, therefore, the following
rule :—

To the constant log, .... 1-7169

Add log tan mid. lat.,

Add log departure in miles, .

The sum is log of the approximate num-
ber of seconds in the convergence,

Thus for a departure of 1 mile in latitude 20°, the
convergence is 19" only, but in latitude 40° it is 44",
and in latitude 60° it is as much as 90".

It thus appears that the convergence increases very
rapidly in high latitudes, and that in latitude 60° the
bearing of a straight line one mile long and running
approximately E. and W. will at one extremity be different
by 1-5 minutes from what it is at the other.

The amount of convergence is such that when a straight
line is run several miles in length the bearing of the line
as determined by astronomical observation will differ
appreciably at each end. The nearer the place is to the
equator, the longer the line will have to be before the
difference is sufficient to directly observe. In latitude

240 ASTRONOMY FOR SURVEYORS.

40° it is readily observable at the end of an East and West
line two miles long, in latitude 60° the line need be only
one mile long for the difference to be just as readily
detected. There is no such effect in lines running directly
N. and S., as such lines form a part of a meridian of longi-
tude, and the convergence is greatest at the extremities
of lines of given length, when the direction is E. and W.

The investigation we have given for convergence is of
course an approximate one only, and the formulae ob-
tained are not exact, because the earth is not in reality
a true sphere as has been assumed. The results obtained,
however, are quite sufficiently accurate for all but the
most refined geodetic work.

MISCELLANEOUS EXAMPLES.

1. At what height would a signal need to be erected at station B to be
visible from the instrument at A, so that the line of sight would be 10 feet
clear of the summit of an intervening hill at C ?

Height of instrument above sea level at A, 488 feet. Station B, 20 miles
distant from A, 5-2 feet. The summit of the intervening hill, 12 miles from
A, 442 feet.

AIM. 32-7 feet.

2. A man on a height near Pietermaritzburg, 42 miles from Durban,
owing to the clearness of the air can see a ship 6 miles out at sea. Looking
in the other direction he can see the heights of Drakenburg, which he knows
are 110 miles from him. Find the height of the Drakenburg above the
sea, taking the radius of the earth as 3,960 miles. (Educational Times.)

Ans. Half a mile nearly.

3. From a point in latitude 30° South, longitude 120° East, a line at
right angles to the initial meridian is run Easterly for a distance of 18 miles.
Find the true bearing of the line at its Easterly end, its longitude, and the
bearing and distance to a point in that longitude in the same latitude as
the starting point. Assume the radius of the earth to be 3,960 miles.

Ans. (a) 269° 50' 59".

(6) Longitude, 120° 18' 02".
(c) Due South, -024 mile.

4. On the evening of the 12th April, 1911, the altitude at meridian transit
of the star a Hydrse, North of the Zenith was observed from two hills,

THE CONVERGENCE OF MERIDIANS. 241

A and B, a considerable distance apart. Altitudes of a Virginis, in the
eastern sky, were observed simultaneously from both hills by aid of pre-
arranged signals. Several sets were taken, which, reduced to a mean and
cleared of corrections for refraction and level errors, gave the following
results : — -

At station A the meridian altitude of a Hydrse was 63° 22' 40" and the
altitude of a Virginis was 12° 14' 18".

At station B the meridian altitude of a Hydrse was 63° 44' 40" and the
altitude of a Virginis was 12° 44' 18".

The declination of a Hydrse was 8° 16' 26" S., and the declination of
a Virginis was 10° 42' 0" S., taken from the Nautical Almanac.

Find the distance between the two hills A and B in miles and decimals,
and the true bearing of each station, treating the earth as a sphere having
a radius of 3,008 miles.

Ans. Distance = 44-64 miles.
Bearing of B from A,

55° 31' 04".
Bearing of A from B,
335° 09' 02".

5. What Is, approximately, the spherical excess in a triangle on the
earth's surface, two sides of the triangle being 163,421 feet and 154,599 feet
respectively, and the observed included angle being 60° 05' 12-32"? What
factors do you require for an exact evaluation ?

6. In latitude 45° N. an observer sees a certain star rise in the N.E. If
the observer travels to another place with a slightly different latitude,
show that the change in direction of the same star at rising will be equal
to the change in latitude.

7. Show that all the stars observable from any one place have the same
rate of change in azimuth at rising.

8. Prove that the rate of change in altitude of a star is always greatest,
when the star is in the prime vertical.

2*3

INDEX.

ABBREVIATIONS, 133.
Alidade level, 95.

Allowance of error of, 95.

Almucantar method for determina-
tion of time, 207-211.
Alternating method for determining
local sidereal time, 58.

— for determining local mean

time, 60.
Altitude, 14.

and azimuth, Determination of,

at short interval,
76-79.

having given R.A.,

declination, lati-
tude and time,
71-74.

— of star at elonga-
tion, 107.

— — after star is in known position,
71.

— of celestial pole, 25.

of pole star, for latitude

determinations, 102.

Altitudes, equal, Method of, correc-
tion for sun obser-
vations, 205.

— of, for azimuth, 98.

— of, for time, 202.

— Meridian, Method of, at both

culminations, 155.

— of, for latitude, 149-

152.

Antarctic circle, 25.
Apparent motion of the stars, 10-13.

— solar time, 44, 46.

— times at same instant in

places of different longi-
tude, 50.

Arctic circle, 25.

Aries, First point of, 15, 43.

— of, time of transit,

63-67.

Arrangement of computations, 192,.

193.
Astronomical co-ordinates, 13-20.

— ternio, Synopsis of, 19.
Atmospheric refraction, 83-87.
Autumnal equinox, 43.
Averaging observations, 195-198.
Azimuth, 14.
Altitude and hour angle at,.

107-111.
Method of determination of,.

71-79.

BESSEL'S formula for refraction, 52,

CALCULATION of the time of elonga-
tion, 106.

Cancer, Tropic of, 24, 25. 37.

Capricorn, Tropic of, 24, 25, 38.

Celestial equator, 13, 39.
— sphere, 8.

— To plot position of sun's

centre on, 40.
Changing face, Elimination of error*

by, 90.

Chronometers, for longitude deter-
mination, 220.

Comparison of, 223.

Circles, Great, 1.

- Small, 2.
Circular parts, Napier's rules of,.

3-5.
Circum -elongation observations for

azimuth, 131-133.

Circum-meridian observations for
latitude, 155-160.

— Limits of time for, 160.
of the sun, 160-163.

Coefficient of refraction, 84.
Collimation, Error of, 88.

244

INDEX.

Comparative advantages of co-
ordinate systems, 17.

Comparison of preceding methods,
62.

Computations, Arrangement of. 118,
192, 193.

Convergence of meridians, 236-241.

Co-ordinates, Astronomical, 13-20.

Corrections, Instrumental, 87.

— to observations of altitude and

azimuth, 80-96.

— to sun observations, 119.
Culmination, Computation of time

of, 162.

— Lower, of star, 155.

— Upper, of star, 155.
Culminations, Lunan for longitude

determination, 231.

DATA necessary for computation,

192.

Daylight, Star observations in, 1 12.
Declination, 15, 17.

— circle, 17.

— Computation of sun's, from

Nautical Almanac data, 39,

118, 119.
Degree of longitude, The length of,

23.
Determination of level error of

axis by means of the striding

level, 93.

— of true meridian, 97-148.
Distance between places whose lati-
tudes and longitudes are known,
26-30.

Distances, Lunar, for longitude de-
terminations, 229.
of stars, 8.

EARTH, The, figure of, 30-32.
- its shape, 21, 31, 32.

— its orbit round the sun,

21.

— Zones of, 24.
Ecliptic, 43.

— Obliquity of the, 43.

Effect of an error of collimation,

88.
I Effect of an error in direction of

prime vertical, 167.
Effect of an error in latitude, 111,

112.
Effect of an error in the longitude

of place of observation, 124.
Effect of an error in the measured

altitude, 124.

Effect of an error in the sun's de-
clination upon the calculated

azimuth, 122-124.
Elimination of instrumental errors

by changing face, 90.
Ellipsoid, The earth an, 32.
Elongation, Altitude, azimuth, and
hour angle at, 107.

— Calculation of time at, 106.

— Observation of star at, 107-111

— Observation of star near, 109.
Equal altitudes, Method of. for

azimuth, 98.
of, for time, 202.

uation of time, 47.
Personal, 224.

|Eq

Equator, Celestial, 13, 39.
| Equinoctial points, 43.
! Equinoxes, 38, 39, 43.
Error of transverse axis, Measure-
ment of, 91.
i Errors, Instrumental, 90-96.

— Elimination of, by chang-
ing face, 90.
Excess, Spherical, 7.
Extra meridian observations, Best

time for, 126.
-for azimuth, 113-125.
- for time, 190-207.
— on sun or star, 113-
118, 188.

FIGURE of earth, 31.

First point of Aries, 15, 43.

- Time of transit of, 63-67.

First point of Libra, 43.

Flash light signals for longitude
determinations, 223.

Formula, Choice of, for extra-
meridian observations, 191.

Frigid Zone, 25.

INDEX.

245

GEOCENTRIC latitude, 32-34.
Geographical latitude, 32-34.
Gnomon, 212.
Great circles, 1.

HORIZON, Celestial, 10.
Horizontal dial, 212-214.

lax, 82.

Hour angle, 18.

— being known, to deter-
mine time, 18, 193.
— of star at elongation, 18.

INCLINATION of earth's axis to plane
of orbit, 36-38.

LATITUDE by altitude of pole star,
169-172.

— by circum-meridian observa-

tions, 155.
by horizontal angle between

two circumpolar stars at

elongation, 175.
• by meridian altitudes, 149-155.

— by prime vertical transits, 163.
by rate of change of altitude

near prime vertical, 163.

- Effect of an error in, 111-113,

198-200.

— Geocentric, 32-34.

— Geographical, 32-34.

— Methods of determination of,

149-181.

- Terrestrial, 21.

Length of a degree of longitude, 23.
Libra, First point of, 43.
Local mean time, 49, 55, 70.

— sidereal time, 50-54, 70.
Longitude by comparison of chrono-
meters, 223.

— by electric telegraph, 221.

— by flash-light signals, 227.

1 Longitude by lunar culminations,
231.

— by lunar distances, 229.

— by lunar occultations, 233.

— by personal equation, 224.

— by portable chronometers, 220.

— by recording and receiving

signals, 223.

— Length of a degree of, 23.

; • Methods of determination of,

219-235.

- Terrestrial, 21.
Lunar observations for longitude,

228-234

MEAN noon, 47.

— solar time, 46, 47.

— time, reduction to sidereal,

52.
Meridian, 14.

— altitudes for determination of

latitude, 149-152.

— altitudes of a star at both lower

and upper culminations, 155.

— Determination of, by circum-

elongation observations,
131-133.

— of, by close circumpolar

star, 127.

— of, by equal altitudes, 98.
of, by extra-meridian

observations 113, 115,
116, 190.

— of, by star at elongation,

102.

— of true, 97-107.

transit, Computation of time
of, 162.

— transits on both sides of Zenith.

187.

i Meridians, Convergence of, 236-241.
Methods of determining latitude,

149-181.

— longitude, 219-235.
time, 182-218.

— true meridian, 97-148.
Motion, Apparent, of stars, 10-13.

— of sun, 35-38.

-- in right ascension and declin-
ation, 30.
Moon's motior , 228-234.

246

INDEX.

N : Right-angled spherical

NADIR, 9. Solution of, 3.

Napier's rules of circular parts, 3-5. Right ascension, 15, 16.
Nautical almanac, 39, 47.

data with regard to time,

67-70.

triangles,

Noon, Apparent, 47.

Mean, 47.

North temperate zone, 25.
Notation, 133.

OBLATE spheroid, The earth an, 31.
Oblique-angled spherical triangles,

Solution of, 5-7.
Oblique dial, 215.
Obliquity of ecliptic, 43.
Observations on both East and

West stars, 195-198.
Occultations, Lunar, for longitude

determination, 233.

PARALLAX, 80-83.

Horizontal, 82.

Parallel of latitude, 23.
Personal equation, 224.
Plot position of sun's centre on

celestial sphere, 40-42.
Polar distance, 17,
Pole, Celestial, 25.
Portable chronometers, 220.
Prime vertical, 18, 163.

dial, 214.

transits, for determina-
tion of latitude, 172.

RECORDING and receiving telegraphic
signals for longitude, 223.

Referring mark for azimuth obser-
vations, 97-106.

Refraction, atmospheric, Correction
for, 83.

Relative accuracy of methods for
finding longitude, 234.

Residual instrumental errors, 87.

and declination, Motion
in, 39.

of star, To determine, 15,

74, 75.
Rising of a celestial body, Time of,

215.
Rules of circular parts, 3-5.

SETTING of a celestial body, Time

of, 215.
Sidereal day, 17.

— time, 17, 44, 46.

— at local mean noon, 54.

— reduction to mean time,

60-62.

Small circles, 2.
Solar time, Apparent, 44.
Solstitial points, 43.
Solution of spherical triangles, 3-7.
South temperate zone, 25.
Sphere, Celestial, 8.
Spherical excess, 7.

triangles, Oblique, 5-7.

Right-angled, 2-5.

Standard time, 51.

— to change to local mean

time, 52.

Star observations in daylight, 112.
Stars, Apparent motion of, 10-13.

averaging several observations

of the same, 195-198.

— observations on both East and

West, 195-198.
Stile of sun dial, 212.
Striding level, Correction to prime

vertical observations,

168.
Correction to time of

meridian transit, 129.

- Use of, 93.

Sun, Apparent motion of, 35-38.
-dials, 211.

- Horizontal, 212.

— Oblique, 215.

- Vertical, 214.

— Earth's orbit round, 36.

— Motion in R.A. and declination

of, 39.

INDEX.

247

Sun, observations, 116-118.

Semi-diameter of, 39.

Sun's apparent annual path, 42,43.

motion among the stars,

35, 36.

— centre in celestial sphere, To

plot position of, 40-42.

— declination, Computation of,

39, 118, 119.
Synopsis of astronomical terms, 19.

TABLES for reduction of circum-
meridian observations, 437, 176,
177.

Telegraphic signals for longitude
determination, 223.

Temperate Zone, 25.

Terrestrial latitude and longitude,
21-23.

Three systems of time measurement,
46-47.

Time, Apparent solar, 44, 46.

determination bv almucantar

method, 207-211.

— by equal altitude obser-

vations, 202.

— by extra meridian obser-

vations, 190-207.

by meridian transits, 62,

182.

— Equation of, 47.

— Local mean, 49, 55-70.

- - reduction to sider-
eal, 52.

— sidereal, 50, 54-70.

— reduction' to mean,

52.
Mean, 46.

Time measurement, Three systems
of, 46, 47.

— Nautical almanac data with

regard to, 67-70.

— observations upon a close cir-

cumpolar star, 127-131.

— of meridian transit of star, 162.

— of rising or setting of celestial

body, 215.

of transit of first point of

Aries, 63-67.

— Standard, of different countries,

51.

Torrid Zone, 25.
Transit, meridian, Time of, 182.

— of first point of Aries, 63-67.
Transits, Meridian, on both sides

of Zenith, 187.
Transverse axis, error due to want

of horizontality, 185.
Triangles, spherical. Solution of,

3.
Tropic of Cancer, 24, 25, 37.

— of Capricorn, 24, 25, 38.

True meridian, Determination of,

97.
Tycho Brahe and the sun's position

among the stars, 36.

VERNAL equinox, 43

ZENITH, 9.

pair observations of stars for

latitude, 152-154.
Zone, North temperate, 25.
South temperate, 25.

Zones of the earth, 24.

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